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example? Explain why or why not. Developing Skills In 3β12, y varies directly as x. In each case: a. What is the constant of variation? b. Write an equation for y in terms of x. c. Using an appropriate scale, draw the graph of the equation written in part b. d. What is the slope of the line drawn in part c? 3. The perimeter of a square (y) is 12 centimeters when the length of a side of the square (x) is 3 centimeters. 4. Jeanne can type 90 words (y) in 2 minutes (x). 5. A printer can type 160 characters (y) in 10 seconds (x). Graphing Direct Variation 377 6. A cake recipe uses 2 cups of flour (y) to 11 2 7. The length of a photograph (y) is 12 centimeters when the length of the negative from cups of sugar (x). which it is developed (x) is 1.2 centimeters. 8. There are 20 slices (y) in 12 ounces of bread (x). 9. Three pounds of meat (y) will serve 15 people (x). 10. Twelve slices of cheese (y) weigh 8 ounces (x). 11. Willie averages 3 hits (y) for every 12 times at bat (x). 12. There are about 20 calories (y) in three crackers (x). Applying Skills 13. If a car travels at a constant rate of speed, the distance that it travels varies directly as time. If a car travels 75 miles in 2.5 hours, it will travel 110 feet in 2.5 seconds. a. Find the constant of variation in miles per hour. b. Draw a graph that compares the distance that the car travels in miles to the number of hours traveled. Let the horizontal axis represent hours and the vertical axis represent distance. c. Find the constant of variation in feet per second. d. Draw a graph that compares the distance the car travels in feet to the number of sec- onds traveled. 14. In typing class, Russ completed a speed test in which he typed 420 characters in 2 minutes. His teacher told him to let 5 characters equal 1 word. a. Find Russβs rate on the test in characters per second. b. Draw a graph that compares the number of characters Russ typed to the number of minutes that he typed. (Let the horizontal axis represent minutes and the vertical axis represent characters.) c. Find Russβs rate |
on the test in words per minute. d. Draw a graph that compares the number of words Russ typed to the number of min- utes that he typed. (Let the horizontal axis represent minutes and the vertical axis represent words.) In 15β20, determine if the two variables are directly proportional. If so, write the equation of variation. 15. Perimeter of a square (P) and the length of a side (s). 16. Volume of a sphere (V) and radius (r). 17. The total amount tucked away in a piggy bank (s) and weekly savings (w), starting with an initial balance of $50. 378 Graphing Linear Functions and Relations 18. Simple interest on an investment (I) in one year and the amount invested (P), at a rate of 2.5%. 19. Length measured in centimeters (c) and length measured in inches (i). 20. Total distance traveled (d) in one hour and average speed (s). 9-9 GRAPHING FIRST-DEGREE INEQUALITIES IN TWO VARIABLES When a line is graphed in the coordinate plane, the line is a plane divider because it separates the plane into two regions called half-planes. One of these regions is a half-plane on one side of the line; the other is a half-plane on the other side of the line. Let us consider, for example, the horizontal line y 3 as a plane divider. As shown in the graph below, the line y 3 and the two half-planes that it forms determine three sets of points: 1. The half-plane above the line y 3 is the set of all points whose y-coordinates are greater than 3, that is, y 3. For example, at point A, y 5; at point B, y 4. 2. The line y 3 is the set of all points whose y-coordinates are equal to 3. For example, y 3 at each point C(4, 3), D(0, 3), and G(6, 3). 3. The half-plane below the line y 3 is the set of all points whose y-coordinates are less than 3, that is, y 3. For example, at point E, y 1; at point F, y 2. y (β2, 4)B A(3, 5) y = 3 C(β4, 3) D(0, 3) |
(6, 3)G (β3, 1)E 1 β1 β1 O 1 F(0, β2) x Together, the three sets of points form the entire plane. To graph an inequality in the coordinate plane, we proceed as follows: 1. On the plane, represent the plane divider, for example, y 3, by a dashed line to show that this divider does not belong to the graph of the half-plane. y 3 y = 3 1 β1 β1 O 1 x Graphing First-Degree Inequalities in Two Variables 379 2. Shade the region of the half-plane whose points satisfy the inequality. To graph y 3, shade the region above the plane divider. 3. To graph y 3, shade the region below the plane divider. y 3 1 β1 β1 y > 3 y = 3 O 1 Graph of y > 3 y 3 1 β1 β1 y = 3 y < 3 O 1 Graph of y < 3 x x Let us consider another example, where the plane divider is not a horizontal line. To graph the inequality y 2x or the inequality y 2x, we use a dashed line to indicate that the line y 2x is not a part of the graph. This dashed line acts as a boundary line for the half-plane being graphed. y (1, 4)B y = 2x y > 2x O x y y = 2x y < 2x O x C (1, β1) Graph of y > 2x Graph of y < 2x The graph of y 2x is the shaded half-plane above the line y 2x. It is the set of all points in which the y-coordinate is greater than twice the x-coordinate. The graph of y 2x is the shaded half-plane below the line y 2x. It is the set of all points in which the y-coordinate is less than twice the x-coordinate. 380 Graphing Linear Functions and Relations An open sentence such as y 2x means y 2x or y 2x. The graph of y 2x is the union of two disjoint sets. It includes all of the points in the solution set of the inequality y 2x and all of the points in the solution set of the equality y 2x. To indicate that y 2x is part of the graph of y 2x, we draw the graph of y 2 |
x as a solid line. Then we shade the region above the line to include the points for which y 2x. y y > 2x y = 2x O x Graph of y > 2x When the equation of a line is written in the form y mx b, the halfplane above the line is the graph of y mx b and the half-plane below the line is the graph of y, mx 1 b. To check whether the correct half-plane has been chosen as the graph of a linear inequality, we select any point in that half-plane. If the selected point satisfies the inequality, every point in that half-plane satisfies the inequality. On the other hand, if the point chosen does not satisfy the inequality, then the other half-plane is the graph of the inequality. EXAMPLE 1 Graph the inequality y 2x 2. Solution How to Proceed (1) Transform the inequality into one having y as the left member: (2) Graph the plane divider, y 2x 2, by using the y-intercept, 2, to locate the first point (0, 2) on the y-axis. Then use the slope, 2 or, to find other points by moving up 2 and to the right 1: 2 1 (3) Shade the half-plane above the line: This region and the line are the required graph. The half-plane is the graph of y 2 2x. 2, and the line is the graph of y 2 2x 5 2. Note that the line is now drawn solid to show that it is part of the graph. y 2x 2 y 2x 2 y 2 y β 2x = 2 O β2 2 x β2 y y β 2x > 2 2 y β 2x = 2 O β2 2 x β2 Graphing First-Degree Inequalities in Two Variables 381 (4) Check the solution. Choose any point in the half-plane selected as the solution to see whether it satisfies the original inequality, y 2x 2: Select point (0, 5) which is in the shaded region. y 2x 2? 5 2 2(0) $ 2 5 2 β The above graph is the graph of y 2 2x $ 2. EXAMPLE 2 Graph each of the following inequalities in the coordinate plane. a. x 1 b. x 1 c. y 1 d. y 1 Answers y 12 = x x > 1 x < 1 |
β2 O x 2 β2 y 2 O β2 y 2 1 = x x 2 β2 O β2 y 2 O β. x < 1 c. y > 1 d. y < 1 β2 a. x > 1 EXERCISES Writing About Mathematics 1. Brittany said that the graph of 2x y 5 is the region above the line that is the graph of 2x y 5. Do you agree with Brittany? Explain why or why not. 2. Brian said that the union of the graph of x 2 and the graph of x 2 consists of every point in the coordinate plane. Do you agree with Brian? Explain why or why not. Developing Skills In 3β8, transform each sentence into one whose left member is y. 3. y 2x 0 6. 2x y 0 4. 5x 2y 7. 3x y 4 5. y x 3 8. 4y 3x 12 382 Graphing Linear Functions and Relations In 9β23, graph each sentence in the coordinate plane. 9. x 4 12. y 3 15. x y 3 18. 2y 6x 0 21. 9 x 3y 10. x 2 13. y x 2 16. x y 1 19. 2x 3y 6 y x 5 # 1 2 1 22. 11. y 5 y $ 1 2x 1 3 14. 17. x β 2y 4 2y. 1 1 20. 3x 1 1 23. 2x 2y 6 0 In 24β26: a. Write each verbal sentence as an open sentence. b. Graph each open sentence in the coordinate plane. 24. The y-coordinate of a point is equal to or greater than 3 more than the x-coordinate. 25. The sum of the x-coordinate and the y-coordinate of a point is less than or equal to 5. 26. The y-coordinate of a point decreased by 3 times the x-coordinate is greater than or equal to 2. Applying Skills In 27β31: a. Write each verbal sentence as an open sentence. b. Graph each open sentence in the c. Choose one pair of coordinates that could be reasonable values for x and y. coordinate plane. 27. The length of Mrs. Gaugerβs garden (y) is greater than the width (x). 28. The cost of a shirt (y) is less than half the cost of a pair of shoes (x). 29. |
The distance to school (y) is at least 2 miles more than the distance to the library (x). 30. The height of the flagpole (y) is at most 4 feet more than the height of the oak tree (x) nearby. 31. At the water park, the cost of a hamburger (x) plus the cost of a can of soda (y) is greater than 5 dollars. 9-10 GRAPHS INVOLVING ABSOLUTE VALUE To draw the graph of the equation y |x|, we can choose values of x and then find the corresponding values of y. Let us consider the possible choices for x and the resulting y-values: 1. Choose x 0. Since the absolute value of 0 is 0, y will be 0. 2. Choose x as any positive number. Since the absolute value of any positive number is that positive number, y will have the same value as x. For example, if x 5, then y 5 5. Graphs Involving Absolute Value 383 3. Choose x as any negative number. Since the absolute value of any nega- tive number is positive, y will be the opposite of x. For example, if x 3, then y 3 3. Thus, we conclude that x can be 0, positive, or negative, but y will be only 0 or positive. Here is a table of values and the corresponding graphx (β5, 5) (5, 5) (β3, 3) (3, 3) (β1, 1) (1, 1) O 1 β1 β1 x Notice that for positive values of x, the graph of y x is the same as the graph of y x. For negative values of x, the graph of y x is the same as the graph of y x. y 1 y 1 y 1 β1 β1 O 1 x O 1 β1 β1 x O 1 β1 β It should be noted that for all values of x, y = x results in a negative value for y. Therefore, for positive values of x, the graph of y = x is the same as the graph of y x. For negative values of x, the graph of y x is the same as the graph of y x. 384 Graphing Linear Functions and Relations EXAMPLE 1 Draw the graph of y x 2. Solution (1) Make a table of values2 2 β2) Plot the points and draw rays to connect the points that |
were graphed: y (β4, 6) y = x + 2 (5, 7) (3, 5) (β2, 4) (β1, 3) (1, 3) (0, 2) x Calculator Solution (1) Enter the equation into Y1: (2) Graph to the standard window: ENTER: Y MATH ENTER ENTER: ZOOM 6 X,T,,n ) 2 DISPLAY DISPLAY: EXAMPLE 2 Draw the graph of x y 3. Solution By the definition of absolute value, x x and y y. β’ Since (1, 2) is a solution, (1, 2), (1, 2) and (1, 2) are solutions. β’ Since (2, 1) is a solution, (2, 1), (2, 1) and (2, 1) are solutions. β’ Since (0, 3) is a solution, (0, 3) is a solution. β’ Since (3, 0) is a solution, (3, 0) is a solution. Graphs Involving Absolute Value 385 y 1 β1 β1 O 1 x Plot the points that are solutions, and draw the line segments joining them. x + y = 3 Translating, Reflecting, and Scaling Graphs of Absolute Value Functions Just as linear functions can be translated, reflected, or scaled, graphs of absolute value functions can also be manipulated by working with the graph of the absolute value function y 5 ZxZ. For instance, the graph of y x 5 is the graph of y x shifted 5 units down. The graph of y βx is the graph of y x reflected in the x-axis. The graph of y 2x is the graph of y x stretched vertically by a factor of 2, is the graph of y x compressed vertically by a facwhile the graph of 1 tor of. 2 y 5 1 2 ZxZ = x1 2 x Translation Rules for Absolute Value Functions If c is positive: The graph of y x c is the graph of y x shifted c units up. The graph of y x c is the graph of y x shifted c units down. 386 Graphing Linear Functions and Relations For absolute value functions, there are two additional translations that can be done to the graph of y x, horizontal shifting to left or to the right. If c is positive: The graph of y x c is the graph of y x shifted c units to the left. The graph of |
y x c is the graph of y x shifted c units to the right. Reflection Rule for Absolute Value Functions The graph of y x is the graph of y x reflected across the x-axis. Scaling Rules for Absolute Value Functions When c 1, the graph of y cx is the graph of y x stretched vertically by a factor of c. When 0 c 1, the graph of y cx is the graph of y x compressed vertically by a factor of c. In aβe, write an equation for the resulting function if the graph of y 5 ZxZ is: a. shifted 2.5 units down b. shifted 6 units to the right c. stretched vertically by a factor of 3 and shifted 5 units up d. compressed vertically by a factor of and reflected in the x-axis 1 3 e. reflected in the x-axis, shifted 1 unit up, and shifted 1 unit to the left EXAMPLE 3 Solution a. y x 2.5 Answer b. y x 6 Answer c. First, stretch vertically by a factor of 3: Then, translate the resulting function 5 units up: 1 d. First, compress vertically by a factor of : 3 Then, reflect in the x-axis: e. First, reflect in the x-axis: Then, translate the resulting function 1 unit up: Finally, translate the resulting function 1 unit to the left: y 3x y 3x 5 Answer y Answer 1 3 ZxZ 21 y 3 ZxZ y x y x 1 y x 1 1 Answer Graphs Involving Exponential Functions 387 EXERCISES Writing About Mathematics 1. Charity said that the graph of y 2x 1 is the graph of y 2x 1 for x 0 and the graph of 2x 1 for x 0. Do you agree with Charity? Explain why or why not. 2. April said that x y 5 is a function. Do you agree with April? Explain why or why not. 3. Euclid said that, for positive values of c, the graph of y cx is the same as the graph of y cx. Do you agree with Euclid? If so, prove Euclidβs statement. Developing Skills In 4β15, graph each equation. 4. y x 1 8. y 2x 5. y x 3 9. y 2x 1 12. y x 13. x y 4 6. y x 1 10. x y 5 14. Zx Z 2 1 Zy Z 4 5 1 7. y |
x 3 11. x 2y 7 15. 2x 4y 6 0 In 16β19, describe the translation, reflection, and/or scaling that must be applied to y x to obtain the graph of each given function. 17. y 19. y x 1.5 4 18. y x 2 3 16. y βx 4 22ZxZ 1 2 9-11 GRAPHS INVOLVING EXPONENTIAL FUNCTIONS A piece of paper is one layer thick. If we place another piece of paper on top of it, the stack is two layers thick. If we place another piece of paper on top, the stack is now three layers thick. As the process continues, we can describe the number of layers, y, in terms of the number of sheets added, x, by the linear function y 1 x. For example, after we have added seven pieces of paper, the stack is eight layers thick. This is an example of linear growth because the change can be described by a linear function. Now consider another experiment. A piece of paper is one layer thick. If the paper is folded in half, the stack is two layers thick. If the stack is folded again, the stack is four layers thick. If the stack is folded a third time, it is eight layers thick. 388 Graphing Linear Functions and Relations 0 folds 1 fold 2 folds 3 folds 1 layer 2 layers 4 layers 8 layers Although we will reach a point at which it is impossible to fold the paper, imagine that this process can continue. We can describe the number of layers of paper, y, in terms of the number of folds, x, by an exponential function, y 2x. The table at the right shows some values for this function. Such functions are said to be nonlinear. After five folds, the stack is 32 layers thick. After ten folds, the stack would be 210 or 1,024 layers thick. In other words, it would be thinker than this book. This is an example of exponential growth because the growth is described by an exponential function y bx. When the base b is a positive number greater than 1 (b 1), y increases as x increases. x 0 1 2 3 4 5 2x 20 21 22 23 24 25 y 2x 1 2 4 8 16 32 We can draw the graph of the exponential function, adding zero and negative integral values to those given above. Locate the points on the coordinate plane and draw a smooth curve through them. When we draw |
the curve, we are assuming that the domain of the independent variable, x, is the set of real numbers. You will learn about powers that have exponents that are not integers in more advanced math courses. x 0 1 2 3 2x 20 21 22 23 y 2x 2, 4) (1, 2) 1 (0, 1) O 1 2 3 x (β3, )1 8 (β1, )1 2 (β2, )1 4 β3 β2 β1 There are many examples of exponential growth in the world around us. Over an interval, certain populationsβfor example of bacteria, or rabbits, or peopleβgrow exponentially. Graphs Involving Exponential Functions 389 Compound interest is also an example of exponential growth. If a sum of money, called the principal, P, is invested at 4% interest, then after one year the value of the investment, A, is the principal plus the interest. Recall that I Prt. In this case, r 0.04 and t 5 1. A P I = P Prt P(1 rt) P(1+ 0.04(1)) P(1.04) After 2 years, the new principal is (1.04)2P. Replace P by 1.04P in the formula A P(1.04): A 1.04P(1.04) = P(1.04)2 After 3 years, the new principal is (1.04)3P. Replace P by P(1.04)2 in the formula A 5 P(1.04) : A (1.04)2P(1.04) = P(1.04)3 In general, after n years, the value of an investment for which the rate of interest is r can be found by using the formula A P(1 r)n. Note that in this formula for exponential growth, P is the amount present at the beginning; r, a positive number, is the rate of growth; and n is the number of intervals of time during which the growth has been taking place. Since r is a positive number, the base, 1 r, is a number greater than 1. Compare the formula A P(1 r)n with the exponential function y 2x used to determine the number of layers in a stack after x folds. P 1 since we started with 1 layer. The base, (1 r) 2, so r 1 or 100%. |
Doubling the number of layers in the stack is an increase equal to the size of the stack, that is, an increase of 100%. An exponential change can be a decrease as well as an increase. Consider this example. Start with a large piece of paper. If we cut the paper into two equal parts, each part is one-half of the original piece. Now if we cut one of these pieces into two equal parts, each of the parts is one-fourth of the original piece. As this process continues, we can describe the part of the original piece that results from each cut in x terms of the number of cuts. The function B gives the part of the original piece, y, that is the result of x cuts. This is an example of exponential decay. For the exponential function y bx, when the base b is a positive number less than 1 (0 b 1), y decreases as x increases 16 1 32 Some examples of exponential decay include a decrease in population, a fund, or the value of a machine when that decrease can be represented by a constant rate at regular intervals. The radioactive decay of a chemical element such as carbon is also an example of exponential decay. 390 Graphing Linear Functions and Relations Problems of exponential decay can be solved using the same formula as that for exponential growth. In exponential decay, the rate of change is a negative number between 1 and 0 so that the base is a number between 0 and 1. This is illustrated in Example 3. EXAMPLE 1 Draw the graph of y 5 x. 2 3 A B Solution Make a table of values from x 3 to x 3. Plot the points and draw a smooth curve through them 23 5 22 5 21 27 27 y 5 4 3 (β3, )27 8 (β2, ) 9 4 2 (0, 1) 1 (β1, )3 2 (1, )2 3 (2, )4 9 (3, )8 27 β3 β2 β1 O 1 2 3 x Calculator Solution (1) Enter the equation into Y1: (2) Graph to the standard window: ENTER: Y ( 2 3 ) ENTER: ZOOM 6 ^ X,T,,n DISPLAY DISPLAY: Graphs Involving Exponential Functions 391 EXAMPLE 2 A bank is advertising a rate of 5% interest compounded annually. If $2,000 is invested in an account at that rate, find the amount of money in the account after 10 years. Solution This is an example of |
exponential growth. Use the formula A P (1 r)n with r 0.05, P the initial investment, and n 10 years. A 2,000(1 0.05)10 2,000(1.05)10 3,257.78954 Answer The value of the investment is $3,257.79 after 10 years. EXAMPLE 3 The population of a town is decreasing at the rate of 2.5% per year. If the population in the year 2000 was 28,000, what will be the expected population in 2015 if this rate of decrease continues? Give your answer to the nearest thousand. Solution Use the formula for exponential growth or decay. The rate of decrease is entered as a negative number so that the base is a number between 0 and 1. r 0.025 P the initial population n 15 years (x 0 corresponds to the year 2000) A P(1 r)n 28,000(1 (β0.025))15 28,000(0.975)15 19,152.5792 Answer The population will be about 19,000 in 2015. EXERCISES Writing About Mathematics 1. The equation y bx is an exponential function. If the graph of the function is a smooth curve, explain why the value of b cannot be negative. 2. The equation A P(1 r)n can be used for both exponential growth and exponential decay when r represents the percent of increase or decrease. a. How does the value of r that is used in this equation for exponential growth differ from that of exponential decay? b. If the equation represents exponential growth, can the rate be greater than 100%? Explain why or why not. c. If the equation represents exponential decay, can the rate be greater than 100%? Explain why or why not. 392 Graphing Linear Functions and Relations 3. Euler says that the graph of y 3x2 is the graph of y 3x shifted 2 units to the left and that the graph of y 3x 2 is the graph of y 3x shifted 2 units up. Do you agree with Euler? Explain why or why not. Developing Skills In 4β9, sketch each graph using as values of x the integers in the given interval. 4. y 3x, [3, 3] 5. y 4x, [2, 2] 7. y 2 5 x, [3, 3] 8. y x, [2, 2] 1 3 6. y 1 |
.5x, [4, 4] 9. y 5 1.25x, [0, 10] A 10. Compare each of the exponential graphs drawn in exercises 4 through 9. A B B a. What point is common to all of the graphs? b. If the value of the base is greater than 1, in what direction does the graph curve? c. If the value of the base is between 0 and 1, in what direction does the graph curve? 11. a. Sketch the graph of y b. Sketch the graph of 1 2 B y 5 22xA x in the interval [3, 3]. in the interval [3, 3]. c. What do you observe about the graphs drawn in a and b? Applying Skills 12. In 2005, the population of a city was 25,000. The population increased by 20% in each of the next three years. If this rate of increase continues, what will be the population of the city in 2012? 13. Alberto invested $5,000 at 6% interest compounded annually. What will be the value of Albertoβs investment after 8 years? 14. Mrs. Boyko has a trust fund from which she withdraws 5% each year. If the fund has a value of $50,000 this year, what will be the value of the fund after 10 years? 15. Hailey has begun a fitness program. The first week she ran 1 mile every day. Each week she increases the amount that she runs each day by 20%. In week 10, how many miles does she run each day? Give your answer to the nearest mile. 16. Alex received $75 for his birthday. In the first week after his birthday, he spent one-third of the money. In the second week and each of the following weeks, he spent one-third of the money he had left. How much money will Alex have left after 5 weeks? 17. During your summer vacation, you are offered a job at which you can work as many days as you choose. If you work 1 day, you will be paid $0.01. If your work 2 days you will be paid a total of $0.02. If your work 3 days you will be paid a total of $0.04. If you continue to work, your total pay will continue to double each day. a. Would you accept this job if you planned to work 10 days? Explain why or why not. b. Would you accept this job |
if you planned to work 25 days? Explain why or why not. CHAPTER SUMMARY Chapter Summary 393 The solutions of equations or inequalities in two variables are ordered pairs of numbers. The set of points whose coordinates make an equation or inequality true is the graph of that equation or inequality. The graph of a linear function of the form Ax By C is a straight line. The domain of a function is the set of all values the independent variable, the x-coordinate, is allowed to take. This determines the range of a function, that is, the set of all values the dependent variable, the y-coordinate, will take. Set-builder notation provides a mathematically concise method of describing the elements of a set. Roster form lists every element of a set exactly once. A linear function can be written in the form y mx b, where m is the slope and b is the y-intercept of the line that is the graph of the function. A line parallel to the x-axis has a slope of 0, and a line parallel to the y-axis has no slope. The slope of a line is the ratio of the change in the vertical direction to the change in the horizontal direction. If (x1, y1) and (x2, y2) are two points on a line, the slope of the line is slope m y2 2 y1 x2 2 x2. If y varies directly as x, the ratio of y to x is a constant. Direct variation can be represented by a line through the origin whose slope is the constant of variation. The graph of y mx b separates the plane into two half-planes. The halfplane above the graph of y mx b is the graph of y mx b, and the halfplane below the graph of y mx b is the graph of y mx b. The graph of y x is the union of the graph of y x for x 0 and the graph of y x for x 0. The equation y bx, b 0 and b 1, is an example of an exponential function. The exponential equation A P(1 r)n is a formula for exponential growth or decay. A is the amount present after n intervals of time, P is the amount present at time 0, and r is the rate of increase or decrease. In exponential growth, r is a positive number. In exponential decay, r is a negative number in the interval (1, 0). The function y x |
and y x can be translated, reflected, or scaled to graph other linear and absolute value functions. Vertical translations (c 0) Horizontal translations (c 0) Reflection across the x-axis Vertical stretching (c 1) Vertical compression (0 c 1) Linear Function Absolute Value y x c (up) y x c (down) y x c (left) y x c (right) y x y cx y cx y x c (up) y x c (down) y x c (left) y x c (right) y x y cx y cx 394 Graphing Linear Functions and Relations VOCABULARY 9-1 Roster form β’ Set-builder notation β’ Member of a set () β’ Not a member of a set () β’ Relation β’ Domain of a relation β’ Range of a relation β’ Function β’ Domain of a function β’ Range of a function β’ Independent variable β’ Dependent variable 9-2 Graph β’ Standard form β’ Linear equation 9-4 Slope β’ Unit rate of change 9-6 y-intercept β’ x-intercept β’ Intercept form a linear equation β’ Slope-intercept form of a linear equation 9-8 Equation of a direct variation 9-9 Plane divider β’ Half-plane 9-11 Linear growth β’ Exponential function β’ Nonlinear functions β’ Exponential growth β’ Exponential decay REVIEW EXERCISES 1. Determine a. the domain and b. the range of the relation shown in the graph to the right. c. Is the relation a function? Explain why or why not. y 1 O 1 x 2. Draw the graph of x y 5. Is the graph of x y 5 a square? Prove your answer. 3. A function is a set of ordered pairs in which no two ordered pairs have the same first element. Explain why {(x, y) y 2x 1} is not a function. 4. What is the slope of the graph of y 2x 5? 5. What is the slope of the line whose equation is 3x 2y 12? 6. What are the intercepts of the line whose equation is 3x 2y 12? 7. What is the slope of the line that passes through points (4, 5) and (6, 1)? In 8β13, in each case: a. Graph the equation or inequality. b. Find the domain and range of the equation or inequality. 8. y x 2 9. y 3 10. y 2 |
3x Review Exercises 395 11. x 2y 8 12. y x 2 13. 2x y 4 In 14β22, refer to the coordinate graph to answer each question. 14. What is the slope of line k? 15. What is the x-intercept of line k? 16. What is the y-intercept of line k? y 1 O 1 k m x 17. What is the slope of a line that is parallel to line k? 18. What is the slope of a line that is perpendicular to line k? 19. What is the slope of line m? 20. What is the x-intercept of line m? 21. What is the y-intercept of line m? 22. What is the slope of a line that is perpendicular to line m? 23. If point (d, 3) lies on the graph of 3x y 9, what is the value of d? In 24β33, in each case select the numeral preceding the correct answer. 24. Which set of ordered pairs represents a function? (1) {(6, 1), (7, 1), (1, 7), (1, 6)} (2) {(0, 13), (1, 13), (6, 13), (112, 13)} (3) {(3, 3), (3, 4), (3, 4), (3, 3)} (4) {(1, 12), (11, 1), (112, 11), (11, 112)} 25. Which point does not lie on the graph of 3x y 9? (2) (2, 3) (1) (1, 6) (3) (3, 0) (4) (0, 9) 26. Which ordered pair is in the solution set of y 2x 4? (1) (0, 5) (2) (2, 0) (3) (3, 3) (4) (0, 2) 27. Which equation has a graph parallel to the graph of y 5x 2? (l) y 5x (2) y 5x + 3 (3) y 2x (4) y 2x 5 28. The graph of 2x y 8 intersects the x-axis at point (2) (8, 0) 29. What is the slope of the graph of the equation y 4? (3) (0, 4) (1) (0, 8 |
) (4) (4, 0) (1) The line has no slope. (3) 4 (2) 0 (4) 4 30. In which ordered pair is the x-coordinate 3 more than the y-coordinate? (1) (1, 4) (2) (1, 3) (3) (3, 1) (4) (4, 1) 396 Graphing Linear Functions and Relations 31. Which of the following is not a graph of a function? y 1 O (1) (2) (3) (4 32. Using the domain D {4x x the set of integers and 10 4x 20}, what is the range of the function y 2 x 2? (1) {10, 14, 18} (2) {12, 16, 20} (3) {10, 14, 16, 20} (4) {4, 14, 24, 34} 33. Which equation best describes the graph of y |x| reflected across the x-axis, shifted 9 units up, and shifted 2 units to the left? (1) y x 2 9 (2) y = x + 2 9 (3) y x 2 9 (4) y x 2 9 34. a. Plot points A(5, 2), B(3, 3), C(3, 3), and D(5, 2). b. Draw polygon ABCD. c. What kind of polygon is ABCD? d. Find DA and BC. e. Find the length of the altitude from C to DA. f. Find the area of ABCD. Review Exercises 397 In 35β38, graph each equation. 36. y x 2 35. y x 2 37. x 2y 6 38. y 2.5x 39. Tiny Tot Day Care Center has changed its rates. It now charges $350 a week for children who stay at the center between 8:00 A.M. and 5:00 P.M. The center charges an additional $2.00 for each day that the child is not picked up by 5:00 P.M. a. Write an equation for the cost of day care for the week, y, in terms of x, the number of days that the child stays beyond 5:00 P.M. b. Define the slope of the equation found in part a in terms of the infor- mation provided. c. What is the |
charge for 1 week for a child who was picked up at the fol- lowing times: Monday at 5:00, Tuesday at 5:10, Wednesday at 6:00, Thursday at 5:00, and Friday at 5:25? 40. Each time Raphael put gasoline into his car, he recorded the number of gallons of gas he needed to fill the tank and the number of miles driven since the last fill-up. The chart below shows his record for one month. Gallons of Gasoline 12 8.5 10 4.5 Number of Miles 370 260 310 140 13 420 a. Plot points on a graph to represent the information in the chart. Let the horizontal axis represent the number of gallons of gasoline and the vertical axis the number of miles driven. b. Find the average number of gallons of gasoline per fill-up. c. Find the average number of miles driven between fill-ups. d. Locate a point on the graph whose x-coordinate represents the average number of gallons of gasoline per fill-up and whose y-coordinate is the average number of miles driven between fill-ups. e. It is reasonable that (0, 0) is a point on the graph? Draw, through the point that you plotted in d and the point (0, 0), a line that could represent the information in the chart. f. Raphael drove 200 miles since his last fill-up. How many gallons of gasoline should he expect to need to fill the tank based on the line drawn in e? 41. Mandy and Jim are standing 20 feet apart. Each second, they decrease the distance between each other by one-half. a. Write an equation to show their distance apart, D, at s seconds. b. How far apart will they be after 5 seconds? c. According to your equation, will Mandy and Jim ever meet? 398 Graphing Linear Functions and Relations 42. According to the curator of zoology at the Rochester Museum and Science Center, if you count the number of chirps of a tree cricket in 15 seconds and add 40, you will have a close approximation of the actual air temperature in degrees Fahrenheit. To test this statement, Alexa recorded, on several summer evenings, the temperature in degrees Fahrenheit and the number of chirps of a tree cricket in 1 minute. She obtained the following results: Chirps/minute 150 170 140 125 108 118 145 Temperature 80 85 76 71 65 70 76 68 56 95 63 110 67 a. Write an |
equation that states the relationship between the number of chirps per minute of the tree cricket, c, to the temperature in degrees Fahrenheit, F. (To change chirps per minute to chirps in 15 seconds, divide c by 4.) b. Draw a graph of the data given in the table. Record the number of chirps per minute on the horizontal axis, and the temperature on the vertical axis. c. Draw the graph of the equation that you wrote in part a on the same set of axes that you used for part b. d. Can the data that Alexa recorded be represented by the equation that you wrote for part a? Explain your answer. 43. In order to control the deer population in a local park, an environmental group plans to reduce the number of deer by 5% each year. If the deer population is now estimated to be 4,000, how many deer will there be after 8 years? Exploration STEP 1. Draw the graph of y x 2. STEP 2. Let A be the point at which the graph intersects the x-axis, B be any point on the line that has an x-coordinate greater than A, and C be the point at which a vertical line from B intersects the x-axis. STEP 3. Compare the slope of y x 2 with tan BAC. STEP 4. Use a calculator to find mBAC. STEP 5. Repeat Steps 2 through 4 for three other lines that have a positive slope. STEP 6. Draw the graph of y x 2 and repeat Step 2. Let the measure of an acute angle between the x-axis and a line that slants upward be positive and the measure of an acute angle between the x-axis and a line that slants downward be negative. STEP 7. Repeat Steps 3 and 4 for y x 2. STEP 8. Repeat Steps 2 through 4 for three other lines that have a negative slope. STEP 9. What conclusion can you draw? CUMULATIVE REVIEW Part I Cumulative Review 399 CHAPTERS 1β9 Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. 1. The product of 5a3 and 3a4 is (1) 15a7 (2) 15a12 (3) 15a7 (4) 15a12 2. A basketball team won b games and lost 4. The ratio of games won to games played is (1) b 4 (2) b b 1 |
4 (3) 4 b (4) 4 b 1 4 3. In decimal notation, 8.72 10β2 is (1) 87,200 (2) 872 (3) 0.0872 (4) 0.00872 4. Which equation is not an example of direct variation? y x 5 2 (1) y 2x y 5 2 x (2) (3) (4) y 5 x 2 5. The graph of y 2x 4 is parallel to the graph of (1) y 2x 5 (2) 2x y 7 (3) y 2x 3 (4) 2x y 0 6. The area of a triangle is 48.6 square centimeters. The length of the base of the triangle is 3.00 centimeters. The length of the altitude of the triangle is (1) 32.4 centimeters (2) 16.2 centimeters (3) 8.10 centimeters (4) 3.24 centimeters 7. The measure of the larger acute angle of a right triangle is 15 more than twice the measure of the smaller. What is the measure of the larger acute angle? (1) 25 (2) 37.5 (3) 55 (4) 65 8. Solve for x: 1 2(x 4) x. (1) 21 2 (2) 2 (3) 3 (4) 4.5 9. The measure of one leg of a right triangle is 9 and the measure of the hypotenuse is 41. The measure of the other leg is (1) 32 (2) 40 (3) 41.98 (4) 1,762 10. Which of the following geometric figures does not always have a pair of " congruent angles? (1) a parallelogram (2) an isosceles triangle (3) a rhombus (4) a trapezoid 400 Graphing Linear Functions and Relations Part II Answer all questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 11. Cory has planted a rectangular garden. The ratio of the length to the width of the garden is 5 : 7. Cory bought 100 feet of fencing. After enclosing the garden with the fence, he still had 4 feet of fence left. What are the dimensions of his garden? 12 |
. A square, ABCD, has a vertex at A(4, 2). Side AB is parallel to the x-axis and AB 7. What could be the coordinates of the other three vertices? Explain how you know that for the coordinates you selected ABCD is a square. Part III Answer all questions in this part. Each correct answer will receive 3 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 13. A straight line with a slope of 2 contains the point (2, 4). Find the y-coordinate of a point on this line whose x-coordinate is 5. 14. Mrs. Gantrish paid $42 for 8 boxes of file folders. She bought some on sale for $4 a box and the rest at a later time for $6 a box. How many boxes of file folders did she buy on sale? Part IV Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 15. Concentric circles (circles that have the same center) are used to divide the circular face of a dartboard into 4 regions of equal area. If the radius of the board is 12.0 inches, what is the radius of each of the concentric circles? Express your answers to the nearest tenth of an inch. 16. A ramp is to be built from the ground to a doorway that is 4.5 feet above the ground. What will be the length of the ramp if it makes an angle of 22Β° with the ground? Express your answer to the nearest foot. WRITING AND SOLVING SYSTEMS OF LINEAR FUNCTIONS Architects often add an outdoor stairway to a building as a design feature or as an approach to an entrance above ground level. But stairways are an obstacle to persons with disabilities, and most buildings are now approached by means of ramps in addition to or in place of stairways. In designing a ramp, the architect must keep the slant or slope gradual enough to easily accommodate a wheelchair. If the slant is too gradual, however, the ramp may become inconveniently long or may require turns to fit it into the available space.The architect will |
also want to include design features that harmonize with the rest of the building and its surroundings. Solving a problem such as the design of a ramp often involves writing and determining the solution of several equations or inequalities at the same time. CHAPTER 10 CHAPTER TABLE OF CONTENTS 10-1 Writing an Equation Given Slope and One Point 10-2 Writing an Equation Given Two Points 10-3 Writing an Equation Given the Intercepts 10-4 Using a Graph to Solve a System of Linear Equations 10-5 Using Addition to Solve a System of Linear Equations 10-6 Using Substitution to Solve a System of Linear Equations 10-7 Using Systems of Equations to Solve Verbal Problems 10-8 Graphing the Solution Set of a System of Inequalities Chapter Summary Vocabulary Review Exercises Cumulative Review 401 402 Writing and Solving Systems of Linear Functions 10-1 WRITING AN EQUATION GIVEN SLOPE AND ONE POINT You have learned to draw a line using the slope and the coordinates of one point on the line. With this same information, we can also write an equation of a line. EXAMPLE 1 Write an equation of the line that has a slope of 4 and that passes through point (3, 5). Solution METHOD 1. Use the definition of slope. How to Proceed (1) Write the definition of slope: (2) Let (x1, y1) (3, 5) and (x2, y2) (x, y). Substitute these values in the definition of slope: (3) Solve the equation for y in terms of x: slope 4 y2 y1 2 x2 2 x1 y 2 5 x 2 3 y 5 4(x 3) y 5 4x 12 y 4x 7 METHOD 2. Use the slope-intercept form of an equation. How to Proceed (1) In the equation of a line, y = mx b, replace m by the given slope, 4: (2) Since the given point, (3, 5), is on the line, its coordinates satisfy the equation y 4x b. Replace x by 3 and y by 5: (3) Solve the resulting equation to find the value of b, the y-intercept: (4) In y 4x b, replace b by 7: y mx b y 4x b 5 4(3) b 5 12 b β7 b y |
4x 7 Answer y 4x 7 In standard form (Ax By C), the equation of the line is Note: 4x y 7. Writing an Equation Given Slope and One Point 403 EXERCISES Writing About Mathematics 1. Micha says that there is another set of information that can be used to find the equation of a line: the y-intercept and the slope of the line. Jen says that that is the same information as the coordinates of one point and the slope of the line. Do you agree with Jen? Explain why or why not. 2. In Method 1 used to find the equation of a line with a slope of 4 and that passes through point (3, 5): a. How can the slope, 4, be written as a ratio? b. What are the principles used in each step of the solution? Developing Skills In 3β6, in each case, write an equation of the line that has the given slope, m, and that passes through the given point. 3. m 2; (1, 4) 4. m 2; (3, 4) 5. m 3; (2, 1) 6. m 5 25 3 ; (3, 0) 1 2 23 4 7. Write an equation of the line that has a slope of and a y-intercept of 2. and a y-intercept of 0. 8. Write an equation of the line that has a slope of 9. Write an equation of the line, in the form y mx b, that is: a. parallel to the line y 2x 4, and has a y-intercept of 1. b. perpendicular to the line y 2x 4, and has a y-intercept of 1. c. parallel to the line y 3x 6, and has a y-intercept of 2. d. perpendicular to the line y 3x 6, and has a y-intercept of 2. e. parallel to the line 2x 3y 12, and passes through the origin. f. perpendicular to the line 2x 3y 12, and passes through the origin. 10. Write an equation of the line, in the form Ax By C, that is: a. parallel to the line y 4x 1, and passes through point (2, 3). b. perpendicular to the line y 4x 1, and passes through point (2, 3). c. parallel to the line 2y 6x 9, and passes through point |
(2, 1). d. perpendicular to the line 2y 6x 9, and passes through point (2, 1). e. parallel to the line y 4x 3, and has the same y-intercept as the line y 5x 3. f. perpendicular to the line y 4x 3, and has the same y-intercept as the line y 5x 3. 404 Writing and Solving Systems of Linear Functions 11. For aβc, write an equation, in the form y mx b, that describes each graph. a. y 1 1 O 1 Applying Skills y = x + 1 (2, 3) x + y = 5 b. (β4, 1) c. Oβ1 β1 (1, 2) O 12. Follow the directions below to draw a map in the coordinate plane. a. The map begins at the point A(0, 3) with a straight line segment that has a slope of 1. Write an equation for this segment. b. Point B has an x-coordinate of 4 and is a point on the line whose equation you wrote in a. Find the coordinates of B and draw AB. c. Next, write an equation for a segment from point B that has a slope of 21 2. d. Point C has an x-coordinate of 6 and is a point on the line whose equation you wrote in c. Find the coordinates of C and draw BC. 1 e. Finally, write an equation for a line through C that has a slope of. 2 f. Point D lies on the line whose equation you wrote in e and on the y-axis. Does point D coincide with point A? 13. If distance is represented by y and time by x, then the rate at which a car travels along a straight road can be represented as slope. Tom leaves his home and drives for 12 miles to the thruway entrance. On the thruway, he travels 65 miles per hour. a. Write an equation for Tomβs distance from his home in terms of the number of hours that he traveled on the thruway. (Hint: Tom enters the thruway at time 0 when he is at a distance of 12 miles from his home.) b. How far from home is Tom after 3 hours? c. How many hours has Tom driven on the thruway when he is 285 miles from home? 10-2 WRITING AN EQUATION GIVEN TWO POINTS You have learned to |
draw a line using two points on the line. With this same information, we can also write an equation of a line. Writing an Equation Given Two Points 405 EXAMPLE 1 Write an equation of the line that passes through points (2, 5) and (4, 11). Solution METHOD 1. Use the definition of slope and the coordinates of the two points to find the slope of the line. Then find the equation of the line using the slope and the coordinates of one point. How to Proceed (1) Find the slope of the line that passes through the two given points, (2, 5) and (4, 11): (2) In y = mx b, replace m by the slope, 3: (3) Select one point that is on the line, for example, (2, 5). Its coordinates must satisfy the equation y 3x b. Replace x by 2 and y by 5: 2 and y1 4 and y2 5] 11] (2, 5). (4, 11). [x1 [x2 4 2 2 5 6 5 11 2 5 2 5 3 Let P1 Let P2 2 y y 1 2 m 5 x2 2 x1 y mx b y 3x b 5 3(2) b (4) Solve the resulting equation to find the value of b, the y-intercept: 1 b 5 6 b (5) In y 3x b, replace b by 1: y 3x 1 METHOD 2. Let A(2, 5) and B(4, 11) be the two given points on the line and P(x, y) be any point on the line. Use the fact that the slope of PA to write an equation. equals the slope of AB How to Proceed (1) Set slope of PA equal to slope of AB : (2) Solve the resulting equation for y 11 26 22 (x 2 2) y 2 5 5 3x 2 6 y 5 3x 2 1 Check Do the coordinates of the second point, (4, 11), satisfy the equation y 3x 1? 3(4) 1 5? 11 11 11 β Answer y 3x 1 406 Writing and Solving Systems of Linear Functions EXERCISES Writing About Mathematics 1. In step 3 of Method 1, the coordinates (2, 5) were substituted into the equation. Could the coordinates (4, 11), the coordinates of the other point on the line, be substituted instead? Explain your |
answer and show that you are correct. 2. Name the principle used in each step of the solution of the equation in Method 2. Developing Skills In 3β6, in each case write an equation of the line, in the form y mx b, that passes through the given points. 3. (0, 5), (2, 0) 4. (0, 3), (1, 1) 5. (1, 4), (3, 8) 6. (3, 1), (9, 7) In 7β10, in each case write an equation of the line, in the form Ax By C, that passes through the given points. 10. (0, 0), (3, 5) 7. (1, 2), (10, 14) 11. A triangle is determined by the three points A(3, 5), B(6, 4), and C(1, 1). Write the equa- 9. (2, 5), (1, 2) 8. (0, 1), (6, 8) tion of each line in the form y mx b: g CA g AB g BC b. a. c. d. Is triangle ABC a right triangle? Explain your answer. 1 12. A quadrilateral is determined by the four points W(1, 1), X(4, 6), Y, 2 101 2, Z B A 53 5, 53 5. Is B A the quadrilateral a trapezoid? Explain your answer. Applying Skills 13. Latonya uploads her digital photos to an internet service that archives them onto CDs for a fee per CD plus a fixed amount for postage and handling; that is, the amount for postage and handling is the same no matter how many archive CDs she purchases. Last month Latonya paid $7.00 for two archive CDs, and this month she paid $13.00 for five archive CDs. a. Write two ordered pairs such that the x-coordinate is the number of archive CDs pur- chased and the y-coordinate is the total cost of archiving the photos. b. Write an equation for the total cost, y, for x archive CDs. c. What is the domain of the equation found in b? Write your answer in set-builder notation. d. What is the range of the equation found in b? Write your answer in set-builder notation. Writing an Equation Given the Intercepts 407 14. Every |
repair bill at Chickieβs Service Spot includes a fixed charge for an estimate of the repairs plus an hourly fee for labor. Jack paid $123 for a TV repair that required 3 hours of labor. Nina paid $65 for a DVD player repair that required 1 hour of labor. a. Write two ordered pairs (x, y), where x represents the number of hours of labor and y is the total cost for repairs. b. Write an equation in the form y = mx b that expresses the value of y, the total cost of repairs, in terms of x, the number of hours of labor. c. What is the fixed charge for an estimate at Chickieβs Service Spot? d. What is the hourly fee for labor at Chickieβs? 15. A copying service charges a uniform rate for the first one hundred copies or less and a fee for each additional copy. Nancy Taylor paid $7.00 to make 200 copies and Rosie Barbi paid $9.20 to make 310 copies. a. Write two ordered pairs (x, y), where x represents the number of copies over one hun- dred and y represents the cost of the copies. b. Write an equation in the form y = mx b that expresses the value of y, the total cost of the copies, in terms of x, the number of copies over one hundred. c. What is the cost of the first one hundred copies? d. What is the cost of each additional copy? 10-3 WRITING AN EQUATION GIVEN THE INTERCEPTS The x-intercept and the y-intercept are two points on the graph of a line. If we know these two points, we can graph the line and we can write the equation of the line. For example, to write the equation of the line whose x-intercept is 5 and whose y-intercept is 3, we can use two points. β’ If the x-intercept is 5, one point is (5, 0). β’ If the y-intercept is 3, one point is (0, 3). Let (x, y) be any other point on the line. Set the slope of the line from (x, y) to (5, 0) equal to the slope of the line from (5, 0) to (0, 3). 0 2 (23 Solve the resulting equation for y. 5y 3(x 5) 5y 3x 15 y |
5 3 5x 2 3 408 Writing and Solving Systems of Linear Functions The slope-intercept form of the equation shows that the y-intercept is 3. Recall from Section 9-6 that we can write this equation in intercept form to show both the x-intercept and the y-intercept. Start with the equation: Add 3x to each side: Divide each term by the constant term: 5y 3x 15 3x 5y 15 5y 215 5 215 23x 215 1 215 y x 23 5 1 5 1 When the equation is in this form, x is divided by the x-intercept and y is divided by the y-intercept. In other words, when the equation of a line is written in the form, the x-intercept is a and the y-intercept is b. y x b 5 1 a 1 EXAMPLE 1 Write an equation of the line whose x-intercept is 1 and whose y-intercept is 2. Write the answer in standard form (Ax By C). 3 Solution How to Proceed (1) In y x b 5 1 a 1 2, replace a by 1 and b by : 3 (2) Write this equation in simpler form: x 21 21 1 3y x 2 5 1 21 1 2 3 Note: In the original form of the equation, y is divided by. To divide by the same as to multiply by the reciprocal,. 3 2 2 3 is (3) Write the equation with integral coefficients by multiplying each term of the equation by the product of the denominators, 2: 22 x 21 A B 3y 2 2 2 5 22(1) A B 2x 2 3y 5 22 Answer 2x 3y 2 EXAMPLE 2 Find the x-intercept and y-intercept of the equation x 5y 2. Writing an Equation Given the Intercepts 409 Solution Write the equation in the form tion by 2, the constant term: y x b 5 1 a 1. Divide each member of the equa- x 1 5y 5 2 5y x 2 5 2 2 1 2 5y x 2 1 2 5 1 The number that divides x, 2, is the x-intercept. The variable y is multiplied by 5, which is equivalent to saying that it is divided by the reciprocal,, so the 2 2 y-intercept is. 5 2 5 2 Answer The x-intercept is 2 and the y-intercept is. 5 EXERC |
ISES Writing About Mathematics 1. a. Can the equation y 4 be put into the form cept of the line? Explain your answer. y x b 5 1 a 1 b. Can the equation x 4 be put into the form intercept of the line? Explain your answer. y x b 5 1 a 1 to find the x-intercept and y-inter- to find the x-intercept and y- 2. Can the equation x y 0 be put into the form intercept of the line? Explain your answer. y x b 5 1 a 1 to find the x-intercept and y- Developing Skills In 3β10, find the intercepts of each equation if they exist. 3. x y 5 7. 3x 4y 8 4. x 3y 6 8. 7 x 2y y x b 5 1 a 1 11. Express the slope of in terms of a and b. 5. 5y x 10 9. y 3x 1 6. 2x y 2 0 10. y 4 12. For aβc, use the x- and y-intercepts to write an equation for each graph. a. b. c. O 2 β1.5 O β1 β1 2 31 O 1 1 3 410 Writing and Solving Systems of Linear Functions Applying Skills 13. Triangle ABC is drawn on the coordinate plane. Point A is at (4, 0), point B is at (0, 3), and point C is at the origin. a. What is the equation of g AB written in the form y x b 5 1 a 1? b. What is the length of AB? 14. a. Isosceles right triangle ABC is drawn on the coordinate plane. Write the equation of each side of the triangle if C is the right angle at the origin and the length of each leg is 7. b. Is more than one answer to a possible? Explain your answer. 10-4 USING A GRAPH TO SOLVE A SYSTEM OF LINEAR EQUATIONS Consistent Equations The perimeter of a rectangle is 10 feet. When we let x represent the width of the rectangle and y represent the length, the equation 2x 2y 10 expresses the perimeter of the rectangle. This equation, which can be simplified to x y 5, has infinitely many solutions. Let x the width of the rectangle, and y the length of the rectangle. Perimeter: 2x 2y 10 x y 5 (Simplified) Measures of the |
sides: y x 1 If we also know that the length of the rectangle is 1 foot more than the width, the dimensions of the rectangle can be represented by the equation y x 1. We want both of the equations, x y 5 and y = x 1, to be true for the same pair of numbers. The two equations are called a system of simultaneous equations or a linear system. System of simultaneous equations: x y 5 y x 1 The solution of a system of simultaneous equations in two variables is an ordered pair of numbers that satisfies both equations. The graphs of x y 5 and y x 1, drawn using the same set of axes in a coordinate plane, are shown at the right. The possible solutions of x y 5 are all ordered pairs that are coordinates of the points on the line x y 5. The possible solutions of y x 1 are all ordered pairs that are coordinates of the points on the line y x 1. The coordinates of the point of intersection, (2, 3), are a solution of both equations. The ordered pair (2, 3) is a solution of the system of simultaneous equations. y y = x + 1 (2, 3) x 1 β1 β1 O 1 + y = 5 x Using a Graph to Solve a System of Linear Equations 411 Check: Substitute (2, 3) in both equations: x y 5 2 1 3 5? 3 3 β Since two straight lines can intersect in no more than one point, there is no other ordered pair that is a solution of this system. Therefore x 2 and y 3, or (2, 3), is the solution of the system of equations. The width of the rectangle is 2 feet, and the length of the rectangle is 3 feet. When two lines are graphed in the same coordinate plane on the same set of axes, one and only one of the following three possibilities can occur. The pair of lines will: 1. intersect in one point and have one ordered number pair in common; 2. be parallel and have no ordered number pairs in common; 3. coincide, that is, be the same line with an infinite number of ordered num- ber pairs in common. If a system of linear equations such as x y 5 and y x 1 has at least one common solution, it is called a system of consistent equations. If a system has exactly one solution, it is a system of independent equations. Inconsistent Equations Sometimes, as shown at the right, when two linear equations are graphed |
in a coordinate plane using the same set of axes, the lines are parallel and fail to intersect, as in the case of x y 2 and x y 4. There is no common solution for the system of equations x y 2 and x y 4. It is obvious that there can be no ordered number pair (x, y) such that the sum of those numbers, x y, is both 2 and 4. Since the solution set of the system has no members, it is the empty set. y 1 β1 β If a system of linear equations such as x y 2 and x y 4 has no common solution, it is called a system of inconsistent equations. The graphs of two inconsistent linear equations are lines that have equal slopes or lines that have no slopes. Such lines are parallel. 412 Writing and Solving Systems of Linear Functions Dependent Equations Sometimes, as shown at the right, when two linear equations are graphed in a coordinate plane using the same set of axes, the graphs turn out to be the same line; that is, they coincide. This happens in the case of the equations x y 2 and 2x 2y 4. Every one of the infinite number of solutions of x y 2 is also a solution of 2x 2y 4. Thus, we see that 2x 2y 4 and x y 2 are equivalent equations with identical solutions. We note that, when both sides of the equation 2x 2y 4 are divided by 2, the result is x y 2. y 1 β1 β If a system of two linear equations, for example, x y 2 and 2x 2y 4, is such that every solution of one of the equations is also a solution of the other, it is called a system of dependent equations. The graphs of two dependent linear equations are the same line. Note that a system of dependent equations is considered consistent because there is at least one solution. Procedure To solve a pair of linear equations graphically: 1. Graph one equation in a coordinate plane. 2. Graph the other equation using the same set of coordinate axes. 3. One of three relationships will apply: a. If the graphs intersect in one point, the common solution is the ordered pair of numbers that are the coordinates of the point of intersection of the two graphs.The equations are independent and consistent. b. If the graphs have no points in common, that is, the graphs are parallel, there is no solution.The equations are inconsistent. c. If the graphs have all points in common, |
that is, the graphs are the same line, every point on the line is a solution.The equations are consistent and dependent. 4. Check any solution by verifying that the ordered pair satisfies both equations. EXAMPLE 1 Solve graphically and check: 2x y 8 y x 2 Using a Graph to Solve a System of Linear Equations 413 Solution (1) Graph the first equation in a coordinate plane. Solve the first equation for y: 2x y 8 The y-intercept is 8 and the slope is 2. Start at the point (0, 8) and move down 2 units and to the right 1 unit to determine two other points. Or, choose three values of x and find the corresponding values of y to determine three points. Draw the line that is the graph of 2x y 8: y 2x 8 y (0, 81 O β1 1 (2) Graph the second equation using the same set of coordinate axes. Solve the second equation for y: The y-intercept is 2 and the slope is 1. Start at the point (0, 2) and move up 1 unit and to the right 1 unit to determine two other points. Or, choose three values of x and find the corresponding values of y to determine three points. Draw the line that is the graph of y β x 2 on the same set of coordinate axes: y x 2 y x 2 y (0, 8) P (2, 4) 2 x + y = 8 x (0, 2) y β x = 2 1 β1 O β1 1 (3) The graphs intersect at one point, P(2, 4). (4) Check: Substitute (2, 4) in each equation: 2x y 8 2(2) 1 4 5? 2 2 2 β 414 Writing and Solving Systems of Linear Functions Calculator Solution (1) Solve the equations for y: 2x y 8 y x 2 y 2x 8 y x 2 (2) Enter the equations in the Y menu. ENTER: Y X,T,,n (-) 2 X,T,,n 8 ENTER 2 DISPLAY+ 2 (3) Use ZStandard to display the equations. ENTER: ZOOM 6 DISPLAY: (4) Use TRACE to determine the coordinates of the point of intersection. Use the right and left arrow keys to move along the first equation to what appears to be the point of intersection and note the coordinates, (2, 4). Use the |
up arrow key to change to the second equation. Again note the coordinates of the point of intersection, (2, 4). ENTER: TRACE DISPLAY: Y1=β2X+8 * X=2 Y=4 Answer (2, 4), or x 2, y 4, or the solution set {(2, 4)} Using a Graph to Solve a System of Linear Equations 415 EXERCISES Writing About Mathematics 1. Are there any ordered pairs that satisfy both the equations 2x y 7 and 2x 5 y? Explain your answer. 2. Are there any ordered pairs that satisfy the equation y x 4 but do not satisfy the equa- tion 2y 8 2x? Explain your answer. Developing Skills In 3β22, solve each system of equations graphically, and check. 3. y 2x 5 y x 4 6. x y 5 x 3y 9 9. y 3x 2x y 10 12. 3x y 6 y 3 1 15. y 3 3x 2x y 8 18. 5x 3y 9 5y 13 x 21. y 2x 6 0 y x 4. y 2x 3 1 y 3 2x 7. y 2x 1 x 2y 7 10. x 3y 9 x 3 13. y 2x 4 x y 5 16. 3x y 13 x 6y 7 19. x 0 y 0 22. 7x 4y 7 0 3x 5y 3 0 5. x y 1 x y 7 8. x 2y 12 y 2x 6 11. y x 2 x 2y 4 14. 2x β y 1 x y 1 17. 2x y 9 6x 3y 15 20. x y 2 0 x y 8 In 23β28, in each case: a. Graph both equations. b. State whether the system is consistent and independent, consistent and dependent, or inconsistent. 24. x y 5 23. x y 1 x y 3 26. 2x y 1 2y 4x 2 2x 2y 10 27. y 3x 2 y 3x 2 25. y 2x 1 y 3x 3 28. x 4y 6 x 2 Applying Skills In 29β32, in each case: a. Write a system of two first-degree equations involving the variables x and y that represent the conditions stated in the problem. b. Solve the system graphically. 29. The sum of two numbers is 8. The difference of these numbers |
is 2. Find the numbers. 416 Writing and Solving Systems of Linear Functions 30. The sum of two numbers is 5. The larger number is 7 more than the smaller number. Find the numbers. 31. The perimeter of a rectangle is 12 meters. Its length is twice its width. Find the dimensions of the rectangle. 32. The perimeter of a rectangle is 14 centimeters. Its length is 3 centimeters more than its width. Find the length and the width. 33. a. The U-Drive-It car rental agency rents cars for $50 a day with unlimited free mileage. Write an equation to show the cost of renting a car from U-Drive-It for one day, y, if the car is driven for x miles. b. The Safe Travel car rental agency rents cars for $30 a day plus $0.20 a mile. Write an equation to show the cost of renting a car from Safe Travel for one day, y, if the car is driven for x miles. c. Draw, on the same set of axes, the graphs of the equations written in parts a and b. d. If Greg will drive the car he rents for 200 miles, which agency offers the less expensive car? e. If Sarah will drive the car she rents for 50 miles, which agency offers the less expensive car? f. If Philip finds that the price for both agencies will be the same, how far is he planning to drive the car? 10-5 USING ADDITION TO SOLVE A SYSTEM OF LINEAR EQUATIONS In the preceding section, graphs were used to find solutions of systems of simultaneous equations. Since most of the solutions were integers, the values of x and y were easily read from the graphs. However, it is not always possible to read values accurately from a graph. For example, the graphs of the system of equations 2x y 2 and x y 2 are shown at the right. The solution of this system of equations is not a pair of integers. We could approximate the solution and then determine whether our approximation was correct by checking. However, there are other, more direct methods of solution. Algebraic methods can be used to solve a system of linear equations in two variables. Solutions by these methods often take less time and lead to more accurate results than the graphic method used in Section 4 of this chapter. y x + y = 2 2x β y = 2 1 O β1β1 1 x Using Addition to Solve a System of Linear Equations 417 To solve |
a system of linear equations such as 2x y 2 and x y 2, we make use of the properties of equality to obtain an equation in one variable. When the coefficient of one of the variables in the first equation is the additive inverse of the coefficient of the same variable in the second, that variable can be eliminated by adding corresponding members of the two equations. The system 2x y 2 and x y 2 can be solved by the addition method as follows: Solve for x (1) Since the coefficients of y in the two equations are additive inverses, add the equations to obtain an equation that has only one variable: (2) Solve the resulting equation for x: 2x y 2 x y 2 4 3x x 5 4 3 (3) Replace x by its value in either of the given equations: Solve for y (4) Solve the resulting equation for y 24 Check: Substitute for x and that these values make the given equations true: 2 3 4 3 for y in each of the given equations, and show 2 A 2x 2 y 5 2 4 2 2 3 5? 2 6 3 5? 2 6 3 5? 2 2 5 2 β We were able to add the equations in two variables to obtain an equation in one variable because the coefficients of one of the variables were additive inverses. If the coefficients of neither variable are additive inverses, we can multiply one or both equations by a convenient constant or constants. For clarity, it is often helpful to label the equations in a system. The procedure is shown in the following examples. KEEP IN MIND When solving a system of linear equations for a variable using the addition method, if the result is: β’ the equation 0 0, then the system is dependent; β’ a false statement (such as 0 3), then the system is inconsistent. 418 Writing and Solving Systems of Linear Functions EXAMPLE 1 Solve the system of equations and check: x 3y 13 x y 5 [A] [B] Solution How to Proceed (1) Since the coefficients of the variable x are the same in both equations, write an equation equivalent to equation [B] by multiplying both sides of equation [B] by 1. Now, since the coefficients of x are additive inverses, add the two equations so that the resulting equation involves one variable, y: (2) Solve the resulting equation for the variable y: (3) Replace y by its value in either of the given equations: ( |
4) Solve the resulting equation for the remaining variable, x: x 3y 13 [A] x y 5 1[B] 2y B] Check Substitute 1 for x and 4 for y in each of the given equations to verify the solution: x 3y 13 1 1 3(4) 5? 13 1 1 12 5? 13 13 13 β x y 5 1 1 4 5? 5 5 5 β Answer x 1, y 4 or (1, 4) Note: If equation [A] in Example 1 was x y 13, then the system of equations would be inconsistent. x y 13 [A] x y 5 [B] 0 8 β Using Addition to Solve a System of Linear Equations 419 EXAMPLE 2 Solve the system of equations and check: 5a b 13 4a 3b 18 [A] [B] Solution How to Proceed (1) Multiply both sides of equation [A] by 3 to obtain an equivalent equation, 3[A]: 5a b 13 15a 3b 39 [A] 3[A] Note that the coefficient of b in 3[A] is the additive inverse of the coefficient of b in [B]. (2) Add the corresponding members of equations 3[A] and [B] to eliminate the variable b: (3) Solve the resulting equation for a: (4) Replace a by its value in either of the given equations and solve for b: 3[A] [B] [A] 15a 3b 39 4a 3b 18 57 19a a 3 5a b 13 5(3) b 13 15 b 13 b 2 Check Substitute 3 for a and 2 for b in the given equations: 5a b 13 4a 3b 18 5(3) 1 (22) 5? 13 15 1 (22) 5? 13 13 13 β 4(3) 2 3(22) 5? 18 12 1 6 5? 18 18 18 β Answer a 3, b 2, or (a, b) (3, 2) Order is critical when expressing solutions. While (a, b) (3, 2) is the solution to the above system of equations, (a, b) (2, 3) is not. Be sure that all variables correspond to their correct values in your answers. 420 Writing and Solving Systems of Linear Functions EXAMPLE 3 Solve and check: 7x 5 2y 3y 16 2x |
[A] [B] Solution How to Proceed (1) Transform each of the given equations into equivalent equations in which the terms containing the variables appear on one side and the constant appears on the other side: (2) To eliminate y, find the least common multiple of the coefficients of y in equations [A] and [B]. That least common multiple is 6. We want to write one equation in which the coefficient of y is 6 and the other in which the coefficient of y is 6. Multiply both sides of equation [A] by 3, and multiply both sides of equation [B] by 2 so that the new coefficients of y will be additive inverses, 6 and 6: (3) Add the corresponding members of this last pair of equations to eliminate variable y: (4) Solve the resulting equation for variable x: (5) Replace x by its value in any equation containing both variables: (6) Solve the resulting equation for the remaining variable, y: 7x 5 2y 7x 2y 5 3y 16 2x 2x 3y 16 [A] [B] 3(7x 2y 5) 2(2x 3y 16) 3[A] 2[B] 3[A] 2[B] [B] 21x 6y 15 4x 6y 32 17 17x x 1 3y 16 2x 3y 16 2(1) 3y 16 2 3y 18 y 6 Check Substitute 1 for x and 6 for y in each of the original equations to verify the answer. The check is left to you. Answer x 1, y 6 or (1, 6) Using Addition to Solve a System of Linear Equations 421 EXERCISES Writing About Mathematics 1. Raphael said that the solution to Example 3 could have been found by multiplying equation [A] by 2 and equation [B] by 7. Do you agree with Raphael? Explain why or why not. 2. Fernando said that the solution to Example 3 could have been found by multiplying equation [A] by 3 and equation [B] by 2. Do you agree with Fernando? Explain why or why not. Developing Skills In 3β26, solve each system of equations by using addition to eliminate one of the variables. Check your solution. 3. x y 12 x y 4 6. c 2d 14 c 3d 9 9. β2m 4n 13 6m 4n 9 12. 5x 2y 20 2 |
x 3y 27 15. 5r 2s 8 3r 7s 1 18. 4a 6b 15 6a β 4b 10 21. 3x 4y 2 x 2(7 y) 2a 1 1 1 3 2a 2 4 3b 5 8 3b 5 24 24. 4. a b 13 a b 5 7. a 4b 8 a 2b 0 10. 4x y 10 2x 3y 12 13. 2x y 26 3x 2y 42 16. 3x 7y 2 2x 3y 3 19. 2x y 17 5x 25 y 22. 3x 5(y 2) 1 8y 3x 25. c 2d 1 2 3c 5d 26 5. 3x y 16 2x y 11 8. 8a 5b 9 2a 5b 4 11. 5x 8y 1 3x 4y 1 14. 2x 3y 6 3x 5y 15 17. 4x 3y 1 5x 4y 1 20. 5r 3s 30 2r 12 3s 3x 1 1 1 3x 2 1 1 26. 2a 3b 3a 2 1 2 4y 5 10 2y 5 4 2b 5 2 23. Applying Skills 27. Pepe invested x dollars in a savings account and y dollars in a certificate of deposit. His total investment was $500. After 1 year he received 4% interest on the money in his savings account and 6% interest on the certificate of deposit. His total interest was $26. To find how much he invested at each rate, solve the following system of equations: x y 500 0.04x 0.06y 26 422 Writing and Solving Systems of Linear Functions 28. Mrs. Briggs deposited a total of $400 in two different banks. One bank paid 3% interest and the other paid 5%. In one year, the total interest was $17. Let x be the amount invested at 3% and y be the amount invested at 5%. To find the amount invested in each bank, solve the following system of equations: x y 400 0.03x 0.05y 17 29. Heather deposited a total of $600 in two different banks. One bank paid 3% interest and the other 6%. The interest on the account that pays 3% was $9 more than the interest on the account that pays 6%. Let x be the amount invested at 3% and y be the amount invested at 6%. To find the amount Heather invested |
in each account, solve the following system of equations: x y 600 0.03x 0.06y 9 30. Greta is twice as old as Robin. In 3 years, Robin will be 4 years younger than Greta is now. Let g represent Gretaβs current age, and let r represent Robinβs current age. To find their current ages, solve the following system of equations: g 2r r 3 g 4 31. At the grocery store, Keith buys 3 kiwis and 4 zucchinis for a total of $4.95. A kiwi costs $0.25 more than a zucchini. Let k represent the cost of a kiwi, and let z represent the cost of a zucchini. To find the cost of a kiwi and the cost of a zucchini, solve the following system of equations: 3k 4z 4.95 k z 0.25 32. A 4,000 gallon oil truck is loaded with gasoline and kerosene. The profit on one gallon of gasoline is $0.10 and $0.13 for a gallon of kerosene. Let g represent the number of gallons of gasoline loaded onto the truck and k the number of gallons of kerosene. To find the number of gallons of each fuel that were loaded into the truck when the profit is $430, solve the following system of equations: g k 4,000 0.10g 0.13k 430 10-6 USING SUBSTITUTION TO SOLVE A SYSTEM OF LINEAR EQUATIONS Another algebraic method, called the substitution method, can be used to eliminate one of the variables when solving a system of equations. When we use this method, we apply the substitution principle to transform one of the equations of the system into an equivalent equation that involves only one variable. Using Substitution to Solve a System of Linear Equations 423 Substitution Principle: In any statement of equality, a quantity may be substituted for its equal. To use the substitution method, we must express one variable in terms of the other. Often one of the given equations already expresses one of the variables in terms of the other, as seen in Example 1. EXAMPLE 1 Solve the system of equations and check: 4x 3y 27 y 2x 1 [A] [B] Solution How to Proceed (1) Since in equation [B], both y and 2x 1 name the same number, eliminate y in equation [ |
A] by replacing it with 2x 1: (2) Solve the resulting equation for x: (3) Replace x with its value in any equation involving both variables: (4) Solve the resulting equation for y: 4x 3y 27 y 2x 1 4x 3(2x 1) 27 4x 6x 3 27 10x 3 27 10x 30 x 3 y 2x 1 y 2(3) 1 y 6 1 y 5 [A] [B] [BβA] [B] Check Substitute 3 for x and 5 for y in each of the given equations to verify that the resulting sentences are true. 4x 3y 27 4(3) 1 3(5) 5? 27 12 1 15 5? 27 27 27 β y 2x 1 5 5? 2(3) 2 1 5 5? 6 2 1 5 5 β Answer x 3, y 5 or (3, 5) 424 Writing and Solving Systems of Linear Functions EXAMPLE 2 Solve the system of equations and check: 3x 4y 26 x 2y 2 [A] [B] Solution In neither equation is one of the variables expressed in terms of the other. We will use equation [B] in which the coefficient of x is 1 to solve for x in terms of y. How to Proceed (1) Transform one of the equations into an equivalent equation in which one of the variables is expressed in terms of the other. In equation [B], solve for x in terms of y: (2) Eliminate x from equation [A] by replacing it with its equal, 2 2y, from step (1): (3) Solve the resulting equation for y: (4) Replace y by its value in any equation that has both variables: (5) Solve the resulting equation for x: x 2y 2 x 2 2y [B] 3x 4y 26 3(2 2y) 4y 26 [A] [BβA] 6 6y 4y 26 6 10y 26 10y 20 y 2 x 2 2y x 2 2(2) x 2 4 x 6 [B] Check Substitute 6 for x and 2 for y in each of the given equations to verify that the resulting sentences are true. This check is left to you. Answer x 6, y 2 or (6, 2) EXERCISES Writing About Mathematics 1. In Example 2, the system of equations could have been solved by first |
solving the equation 3x 4y 26 for y. Explain why the method used in Example 2 was easier. 2. Try to solve the system of equations x 2y 5 and y in the first equation. What conclusion can you draw? 1 2x 1 1 by substituting 1 2x 1 1 for y Using Substitution to Solve a System of Linear Equations 425 Developing Skills In 3β14, solve each system of equations by using substitution to eliminate one of the variables. Check. 3. y x x y 14 6. y x 1 x y 9 9. a 2b 2 2a b 5 12. 2x 3y 4x 3y 12 4. y 2x x y 21 7. a 3b 1 5b 2a 1 10. 7x 3y 23 x 2y 13 13. 4y 3x 5x 8y 4 5. a 2b 5a 3b 13 8. a b 11 3a 2b 8 11. 4d 3h 25 3d 12h 9 14. 2x 3y 7 4x 5y 25 In 15β26, solve each system of equations by using any convenient algebraic method. Check. 15. s r 0 r s 6 18. 3x 8y 16 5x 10y 25 21. 3(y 6) 2x 3x 5y 11 24. 3x 4y 3x 1 8 5 5 3y 2 1 2 Applying Skills 16. 3a b 13 2a 3b 16 19. y 3x 3x 1 1 1 2y 5 11 22. x y 300 25. 0.1x 0.3y 78 a 3 5 a 1 b 6 5a 2b 49 17. y x 2 20. 3x y 16 a 2 2 3b 5 4 3 5a 1 b 5 15 23. 3d 13 2c 3c 1 d 2 8 26. a 3(b 1) 0 2(a 1) 2b 16 27. The length of the base of an isosceles triangle is 6 centimeters less than the sum of the lengths of the two congruent sides. The perimeter of the triangle is 78 centimeters. a. If x represents the length of each of the congruent sides and y represents the length of the base, express y in terms of x. b. Write an equation that represents the perimeter in terms of x and y. c. Solve the equations that you wrote in a and b to find the lengths of the |
sides of the triangle. 28. A package of batteries costs $1.16 more than a roll of film. Martina paid $11.48 for 3 rolls of film and a package of batteries. a. Express the cost of a package of batteries, y, in terms of the cost of a roll of film, x. b. Write an equation that expresses the amount that Martina paid for the film and batter- ies in terms of x and y. c. Solve the equations that you wrote in a and b to find what Martina paid for a roll of film and of a package of batteries. 426 Writing and Solving Systems of Linear Functions 29. The cost of the hotdog was $0.30 less than twice the cost of the cola. Jules paid $3.90 for a hotdog and a cola. a. Express the cost of a hotdog, y, in terms of the cost of a cola, x. b. Express what Jules paid for a hotdog and a cola in terms of x and y. c. Solve the equations that you wrote in a and b to find what Jules paid for a hotdog and for a cola. 30. Terri is 12 years older than Jessica. The sum of their ages is 54. a. Express Terriβs age, y, in terms of Jessicaβs age, x. b. Express the sum of their ages in terms of x and y. c. Solve the equations that you wrote in a and b to find Terriβs age and Jessicaβs age. 10-7 USING SYSTEMS OF EQUATIONS TO SOLVE VERBAL PROBLEMS You have previously learned how to solve word problems by using one variable. Frequently, however, a problem can be solved more easily by using two variables rather than one variable. For example, we can use two variables to solve the following problem: The sum of two numbers is 8.6. Three times the larger number decreased by twice the smaller is 6.3. What are the numbers? First, we will represent each number by a different variable: Let x the larger number and y the smaller number. Now use the conditions of the problem to write two equations: x y 8.6 The sum of the numbers is 8.6: Three times the larger decreased by twice the smaller is 6.3: 3x 2y 6.3 Solve the system of equations to |
find the numbers: x y 8.6 β 2(x y) 2(8.6) 2x 2y 17.2 3x 2y 6.3 23.5 5x x 4.7 The two numbers are 4.7 and 3.9. x y 8.6 4.7 y 8.6 y 3.9 Using Systems of Equations to Solve Verbal Problems 427 Procedure To solve a word problem by using a system of two equations involving two variables: 1. Use two different variables to represent the different unknown quantities in the problem. 2. Use formulas or information given in the problem to write a system of two equations in two variables. 3. Solve the system of equations. 4. Use the solution to determine the answer(s) to the problem. 5. Check the answer(s) in the original problem. EXAMPLE 1 The owner of a menβs clothing store bought six belts and eight hats for $140. A week later, at the same prices, he bought nine belts and six hats for $132. Find the price of a belt and the price of a hat. Solution (1) Let b the price, in dollars, of a belt, and h the price, in dollars, of a hat. (2) 6 belts and 8 hats |_____| |______| β β β 8h 6b cost $140. β β 140 9 belts and 6 hats |_____| |______| β β β 6h 9b cost $132. β β 132 (3) The least common multiple of the coefficients of b is 18. To eliminate b, write equivalent equations with 18 and 18 as the coefficients of b. Obtain these equations by multiplying both sides of the first equation by 3 and both members of the second equation by 2. Then add the equations and solve for h. 3(6b 8h) 3(140) β 18b 24h 420 2(9b 6h) 2(132) β 18b 12h 264 12h 156 h 13 428 Writing and Solving Systems of Linear Functions Substitute 13 for h in any equation containing both variables. 6b 8h 140 6b 8(13) 140 6b 104 140 6b 36 b 6 (4) Check: 6 belts and 8 hats cost 6($6) 8($13) $36 $104 $140. β 9 belts and 6 hats cost 9($6) 6($ |
13) $54 $78 $132. β Answer A belt costs $6; a hat costs $13. EXAMPLE 2 When Angelo cashed a check for $170, the bank teller gave him 12 bills, some $20 bills and the rest $10 bills. How many bills of each denomination did Angelo receive? Solution (1) Represent the unknowns using two variables: Let x number of $10 bills, and y number of $20 bills. In this problem, part of the information is in terms of the number of bills (the bank teller gave Angelo 12 bills) and part of the information is in terms of the value of the bills (the check was for $170). (2) Write one equation using the number of bills: Write a second equation using the value of the bills. The value of x $10 bills is l0x, the value of y $20 bills is 20y, and the total value is $170: (3) Solve the system of equations: x y 12 10x 20y 170 10x 20y 170 10(x y) 10(12) β β10x 10y 120 10y 50 5 y x y 12 x 5 12 x 7 (4) Check the number of bills: 7 ten-dollar bills and 5 twenty-dollar bills 12 bills β Check the value of the bills: 7 ten-dollar bills are worth $70 and 5 twenty-dollar bills are worth $100. Total value $70 $100 $170 β Answer Angelo received 7 ten-dollar bills and 5 twenty-dollar bills. Using Systems of Equations to Solve Verbal Problems 429 EXERCISES Writing About Mathematics 1. Midori solved the equations in Example 2 by using the substitution method. Which equation do you think she would have solved for one of the variables? Explain your answer. 2. The following system can be solved by drawing a graph, by using the addition method, or by using the substitution method. x y 1 5x 10y 8 a. Which method do you think is the more efficient way of solving this system of equa- tions? Explain why you chose this method. b. Which method do you think is the less efficient way of solving this system of equa- tions? Explain why you chose this method. Developing Skills In 3β9, solve each problem algebraically, using two variables. 3. The sum of two numbers is 36. Their difference is 24. Find the numbers. 4. The |
sum of two numbers is 74. The larger number is 3 more than the smaller number. Find the numbers. 5. The sum of two numbers is 104. The larger number is 1 less than twice the smaller number. Find the numbers. 6. The difference between two numbers is 25. The larger exceeds 3 times the smaller by 4. Find the numbers. 7. If 5 times the smaller of two numbers is subtracted from twice the larger, the result is 16. If the larger is increased by 3 times the smaller, the result is 63. Find the numbers. 8. One number is 15 more than another. The sum of twice the larger and 3 times the smaller is 182. Find the numbers. 9. The sum of two numbers is 900. When 4% of the larger is added to 7% of the smaller, the sum is 48. Find the numbers. Applying Skills In 10β31, solve each problem algebraically using two variables. 10. The perimeter of a rectangle is 50 centimeters. The length is 9 centimeters more than the width. Find the length and the width of the rectangle. 11. A rectangle has a perimeter of 38 feet. The length is 1 foot less than 3 times the width. Find the dimensions of the rectangle. 12. Two angles are supplementary. The larger angle measures 120Β° more than the smaller. Find the degree measure of each angle. 430 Writing and Solving Systems of Linear Functions 13. Two angles are supplementary. The larger angle measures 15Β° less than twice the smaller. Find the degree measure of each angle. 14. Two angles are complementary. The measure of the larger angle is 30Β° more than the mea- sure of the smaller angle. Find the degree measure of each angle. 15. The measure of the larger of two complementary angles is 6Β° less than twice the measure of the smaller angle. Find the degree measure of each angle. 16. In an isosceles triangle, each base angle measures 30Β° more than the vertex angle. Find the degree measures of the three angles of the triangle. 17. At a snack bar, 3 pretzels and 1 can of soda cost $2.75. Two pretzels and 1 can of soda cost $2.00. Find the cost of a pretzel and the cost of a can of soda. 18. On one day, 4 gardeners and 4 helpers earned $360. On another day, working the same number of hours and at the same rate of pay, 5 gardeners and 6 |
helpers earned $480. Each gardener receives the same pay for a dayβs work and each helper receives the same pay for a dayβs work. How much does a gardener and how much does a helper earn each day? 19. A baseball manager bought 4 bats and 9 balls for $76.50. On another day, she bought 3 bats and 1 dozen balls at the same prices and paid $81.00. How much did she pay for each bat and each ball? 20. Mrs. Black bought 2 pounds of veal and 3 pounds of pork, for which she paid $20.00. Mr. Cook, paying the same prices, paid $11.25 for 1 pound of veal and 2 pounds of pork. Find the price of a pound of veal and the price of a pound of pork. 21. One day, Mrs. Rubero paid $18.70 for 4 kilograms of brown rice and 3 kilograms of basmati rice. Next day, Mrs. Leung paid $13.30 for 3 kilograms of brown rice and 2 kilograms of basmati rice. If the prices were the same on each day, find the price per kilogram for each type of rice. 22. Tickets for a high school dance cost $10 each if purchased in advance of the dance, but $15 each if bought at the door. If 100 tickets were sold and $1,200 was collected, how many tickets were sold in advance and how many were sold at the door? 23. A dealer sold 200 tennis racquets. Some were sold for $33 each, and the rest were sold on sale for $18 each. The total receipts from these sales were $4,800. How many racquets did the dealer sell at $18 each? 24. Mrs. Rinaldo changed a $100 bill in a bank. She received $20 bills and $10 bills. The number of $20 bills was 2 more than the number of $10 bills. How many bills of each kind did she receive? 25. Linda spent $4.50 for stamps to mail packages. Some were 39-cent stamps and the rest were 24-cent stamps. The number of 39-cent stamps was 3 less than the number of 24-cent stamps. How many stamps of each kind did Linda buy? 26. At the Savemore Supermarket, 3 pounds of squash and 2 pounds of eggplant cost $2.85. The cost of 4 pounds of squash and |
5 pounds of eggplant is $5.41. What is the cost of one pound of squash, and what is the cost of one pound of eggplant? 27. One year, Roger Jackson and his wife Wilma together earned $67,000. If Roger earned $4,000 more than Wilma earned that year, how much did each earn? Graphing the Solution Set of a System of Inequalities 431 28. Mrs. Moto invested $1,400, part at 5% and part at 8%. Her total annual income from both investments was $100. Find the amount she invested at each rate. 29. Mr. Stein invested a sum of money in certificates of deposit yielding 4% a year and another sum in bonds yielding 6% a year. In all, he invested $4,000. If his total annual income from the two investments was $188, how much did he invest at each rate? 30. Mr. May invested $21,000, part at 8% and the rest at 6%. If the annual incomes from both investments were equal, find the amount invested at each rate. 31. A dealer has some hard candy worth $2.00 a pound and some worth $3.00 a pound. He makes a mixture of these candies worth $45.00. If he used 10 pounds more of the less expensive candy than he used of the more expensive candy, how many pounds of each kind did he use? 10-8 GRAPHING THE SOLUTION SET OF A SYSTEM OF INEQUALITIES To find the solution set of a system of inequalities, we must find the ordered pairs that satisfy the open sentences of the system. We do this by a graphic procedure that is similar to the method used in finding the solution set of a system of equations. EXAMPLE 1 Graph the solution set of this system: x 2 y 2 Solution (1) Graph x 2 by first graphing the plane divider x 2, which, in the figure at the right, is represented by the dashed line labeled l. The half-plane to the right of this line is the graph of the solution set of x 2. (2) Using the same set of axes, graph y 2 by first graphing the plane divider y 2, which, in the figure at the right, is represented by the dashed line labeled m. The half-plane below this line is the graph of the solution set of y 2. m y l 1 1O 3) The solution |
set of the system x 2 and y 2 consists of the intersection of the solution sets of x 2 and y 2. Therefore, the dark colored region in the lower figure, which is the intersection of the graphs made in steps (1) and (2), is the graph of the solution set of the system x 2 and y 2 432 Writing and Solving Systems of Linear Functions From the graph on page 431, all points in the solution region, and no others, satisfy both inequalities of the system. For example, point (4, 3), which lies in the region, satisfies the system because its x-value satisfies one of the given inequalities, 4 2, and its y-value satisfies the other inequality, 3 2. EXAMPLE 2 Graph the solution set of 3 x 5 in a coordinate plane. Solution The inequality 3 x 5 means 3 x and x 5. This may be written as x 3 and x 5. (1) Graph x 3 by first graphing the plane divider x 3, which, in the figure at the right, is represented by the dashed line labeled l. The half-plane to the right of line x 3 is the graph of the solution set of x 3. (2) Using the same set of axes, graph x 5 by first graphing the plane divider x 5, which, in the figure at the right, is represented by the dashed line labeled m. The half-plane to the left of line x 5 is the graph of the solution set of x 5. (3) The dark colored region, which is the intersection of the graphs made in steps (1) and (2), is the graph of the solution set of 3 x 5, or x 3 and x 5. All points in this region, and no others, satisfy 3 x 5. For example, point (4, 3), which lies in the region and whose x-value is 4, satisfies 3 x 5 because 3 4 5 is a true statement EXAMPLE 3 Graph the following system of inequalities and label the solution set R: x y 4 y 2x 3 Solution (1) Graph x y 4 by first graphing the plane divider x y 4, which, in the figure at the top of page 433, is represented by the solid line labeled l. The line x y 4 and the half-plane above this line together form the graph of the solution set of x y 4. Graphing the Solution Set of a System of Inequalities 433 (2 |
) Using the same set of axes, graph y 2x 3 by first graphing the plane divider y 2x β 3. In the figure at the right, this is the solid line labeled m. The line y 2x 3 and the half-plane below this line together form the graph of the solution set of y 2x 3. (3) The dark colored region, the intersection of the graphs made in steps (1) and (2), is the solution set of the system x y 4 and y 2x 3. Label it R. Any point in the region R, such as (5, 2), will satisfy x y 4 because 5 2 4, or 7 4, is true, and will satisfy y 2x 3 because 2 2(5) 3, or 2 7, is true < 2x β 3 R x y < 2x β 3 1 β1β1 O 1 m EXAMPLE 4 Sandi boards cats and dogs while their owners are away. Each week she can care for no more than 12 animals. For next week she already has reservations for 4 cats and 5 dogs, but she knows those numbers will probably increase. Draw a graph to show the possible numbers of cats and dogs that Sandi might board next week and list all possible numbers of cats and dogs. Solution (1) Let x the number of cats Sandi will board, and y the number of dogs. (2) Use the information in the problem to write inequalities. Sandi can care for no more than 12 animals: She expects at least 4 cats: She expects at least 5 dogs: (3) Draw the graphs of these three y inequalities. (4) The possible numbers of cats and x y 12 x 4 y 5 dogs are represented by the ordered pairs of positive integers that are included in the solution set of the inequalities. These ordered pairs are shown as points on the graph. The points (4, 5), (4, 6), (4, 7), (4, 8), (5, 5), (5, 6), (5, 7), (6, 5), (6, 6), (7, 5), all satisfy the given conditions1β1 O 1 434 Writing and Solving Systems of Linear Functions Answer If there are 4 cats, there can be 5, 6, 7, or 8 dogs. If there are 5 cats, there can be 5, 6, or 7 dogs. If there are 6 cats, there can be 5 or 6 dogs. If there |
are 7 cats, there can be 5 dogs. EXERCISES Writing About Mathematics 1. Write a system of inequalities whose solution set is the unshaded region of the graph drawn in Example 3. Explain your answer. 2. What points on the graph drawn in Example 3 are in the solution set of the system of open sentences x y 4 and y 2x 3? Explain your answer. 3. Describe the solution set of the system of inequalities y 4x and 4x y 3. Explain why this occurs. Developing Skills In 4β18, graph each system of inequalities and label the solution set S. Check one solution. 4. x 1 y 2 7. y x y 2x 3 10. y x 5 y 2x 7 13. x y 3 x β y 6 16. 2x 3y 6 x y 4 0 5. x 0 y 0 8. y 2x y x 3 11. y x 1 x y 2 14 2x x 0 17. 6. y 1 y x 1 9. y 2x 3 y x 12. y 3x 6 y 2x 4 15. 2x y 6 18. x y 2 0 x 1 3 1 3y 2 2 4y # 3 3x, 0 In 19β23, in each case, graph the solution set in a coordinate plane. 19. 1 x 4 22. 2 y 3 20. 5 x 1 23. (y 1) and (y 5) 21. 2 y 6 In 24 and 25, write the system of equation whose solution set is labeled S. Graphing the Solution Set of a System of Inequalities 435 25. y 1 x 1 O 1 x S 24. y 1 O S Applying Skills 26. In Ms. Dwyerβs class, the number of boys is more than twice the number of girls. There are at least 2 girls. There are no more than 12 boys. a. Write the three sentences given above as three inequalities, letting x equal the number of girls and y equal the number of boys. b. On one set of axes, graph the three inequalities written in part a. c. Label the solution set of the system of inequalities S. d. Do the coordinates of every point in the region labeled S represent the possible num- ber of girls and number of boys in Ms. Dwyerβs class? Explain your answer. e. Write one ordered pair that could represent the number of girls and boys in Ms. DwyerοΏ½ |
οΏ½s class. 27. When Mr. Ehmke drives to work, he drives on city streets for part of the trip and on the expressway for the rest of the trip. The total trip is less than 8 miles. He drives at least 1 mile on city streets and at least 2 miles on the expressway. a. Write three inequalities to represent the information given above, letting x equal the number of miles driven on city streets and y equal the number of miles driven on the expressway. b. On one set of axes, graph the three inequalities written in part a. c. Label the solution set of the system of inequalities R. d. Do the coordinates of every point in the region labeled R represent the possible number of miles driven on city streets and driven on the expressway? Explain your answer. e. Write one ordered pair that could represent the number of miles Mr. Ehmke drives on city streets and on the expressway. 28. Mildred bakes cakes and pies for sale. She takes orders for the baked goods but also makes extras to sell in her shop. On any day, she can make a total of no more than 12 cakes and pies. For Monday, she had orders for 4 cakes and 6 pies. Draw a graph and list the possible number of cakes and pies she might make on Monday. 436 Writing and Solving Systems of Linear Functions CHAPTER SUMMARY The equation of a line can be written if one of the following sets of infor- mation is known: 1. The slope and one point 2. The slope and the y-intercept 3. Two points 4. The x-intercept and the y-intercept A system of two linear equations in two variables may be: 1. Consistent: its solution is at least one ordered pair of numbers. (The graphs of the equations are intersecting lines or lines that coincide.) 2. Inconsistent: its solution is the empty set. (The graphs of the equations are parallel with no points in common.) 3. Dependent: its solution is an infinite set of number pairs. (The graphs of the equations are the same line.) 4. Independent: its solution set is exactly one ordered pair of numbers. The solution of an independent linear system in two variables may be found graphically by determining the coordinates of the point of intersection of the graphs or algebraically by using addition or substitution. Systems of linear equations can be used to solve verbal problems. The solution set of a system of inequalities can |
be shown on a graph as the intersection of the solution sets of the inequalities. VOCABULARY 10-4 System of simultaneous equations β’ Linear system β’ System of consistent equations β’ System of independent equations β’ System of inconsistent equations β’ System of dependent equations 10-5 Addition method 10-6 Substitution method β’ Substitution principle 10-8 System of inequalities REVIEW EXERCISES 1. Solve the following system of equations for x and y and check the solution: 5x 2y 22 x 2y 2 2. a. Graph the following system of inequalities and check one solution: Review Exercises 437 y 2x 3 y 2x b. Describe the solution set of the system in terms of the solution sets of the individual inequalities. In 3β8, write the equation of the line, in the form y mx b, that satisfies each of the given conditions. 3. Through (1, 2) with slope 3 1 4. Through (5, 6) with slope 1 2 5. Through (2, 5) and (2, 3) 6. Through (1, 3) and (5, 3) 2 3 7. With slope 2 and y-intercept 0 8. Graphed to the right y 1 1 O x In 9β14, write the equation of the line, in the form of the given conditions. y x b 5 1 a 1, that satisfies each 9. With slope 1 and y-intercept 1 10. With slope 3.5 and y-intercept 1.5 11. With x-intercept 5 and y-intercept 2 3 5 12. With x-intercept and y-intercept 7 4 13. Through (1, 3) and (5, 6) 14. Graphed to the right y 1 O 1 x In 15β17, write the equation of the line, in the form Ax By C, that satisfies each of the given conditions. 15. Parallel to the y-axis and 3 units to the left of the y-axis 16. Through (0, 4) and (2, 0) 17. Through (1, 1) and (4, 5) In 18β20, solve each system of equations graphically and check. 19. y x 18. x y 6 y 2x 6 2x y 3 20. 2y x 4 x y 4 0 438 Writing and Solving Systems of Linear Functions In 21β23, solve each system of equations |
by using addition to eliminate one of the variables. Check. 21. 2x y 10 x y 3 23. 3c d 0 c 4d 52 22. x 4y 1 5x 6y 8 In 24β26, solve each system of equations by using substitution to eliminate one of the variables. Check. 24 x 2y 7 x y 8 25. 3r 2s 20 r 2s 26. x y 7 2x 3y 21 In 27β32, solve each system of equations by using an appropriate algebraic method. Check. 27. x y 0 28. 5a 3b 17 4a 5b 21 31. x y 1,000 0.06x 0.04y t 29. t u 12 1 3u 32. 10t u 24 t u 1 7(10u 1 t) 3x 2y 5 30. 3a 4b 4a 5b 2 33. a. Solve this system of equations algebraically: x y 3 x 3y 9 b. On a set of coordinate axes, graph the system of equations given in part a. In 34β36, graph each system of inequalities and label the solution set A. 34. y 2x 3 y 5 x 35. y 1 2x x 4 36. 2x y 4 x y 2 In 37β41, for each problem, write two equations in two variables and solve algebraically. 37. The sum of two numbers is 7. Their difference is 18. Find the numbers. 38. At a store, 3 notebooks and 2 pencils cost $2.80. At the same prices, 2 notebooks and 5 pencils cost $2.60. Find the cost of a notebook and of a pencil. 39. Two angles are complementary. The larger angle measures 15Β° less than twice the smaller angle. Find the degree measure of each angle. 40. The measure of the vertex angle of an isosceles triangle is 3 times the measure of each of the base angles. Find the measure of each angle of the triangle. 41. At the beginning of the year, Lourdesβs salary increased by 3%. If her weekly salary is now $254.41, what was her weekly salary last year? Cumulative Review 439 Exploration Jason and his brother Robby live 1.25 miles from school. Jason rides his bicycle to school and Robby walks. The graph below shows their relative positions on their way to school one morning. Use the information provided by the graph to describe |
Robbyβs and Jasonβs journeys to school. Who left first? Who arrived at school first? If each boy took the same route to school, when did Jason pass Robby? How fast did Robby walk? How fast did Jason ride his bicycle.25 1.0 0.75 0.50 0.25 0 ROBBY 8:00 8:05 8:10 8:15 8:20 8:25 8:30 TIME JASON CUMULATIVE REVIEW CHAPTERS 1β10 Part I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. 1. The multiplicative inverse of 0.4 is (1) 0.4 (2) 1 (3) 2.5 2. Which of the following numbers has the greatest value? (1) 1.5 (2) 21.5 (3) 2 (4) 2.5 (4) 0.5p 2 " 3. Which of the following is an example of the associative property of addi- tion? (1) 6 (x 3) 6 (3 x) (2) 2(x 2) 2x 4 (3) 3(4a) (3 4)a (4) 3 (4 a) (3 4) a 4. Which of the following is a point on the line whose equation is x y 5? (1) (1, 4) (2) (1, 4) (3) (1, 4) (4) (1, 4) 5. The expression 5 2(3x2 8) is equivalent to (1) 9x2 24 (2) 6x2 21 (3) 9x2 8 (4) 6x2 11 6. The solution of the equation 0.2x 3 x 1 is (2) 5 (1) 5 (3) 0.5 (4) 50 440 Writing and Solving Systems of Linear Functions 7. The product of 3.40 103 and 8.50 102 equals (1) 2.89 100 (4) 2.89 101 8. The measure of A is 12Β° less than twice the measure of its complement. (3) 2.89 102 (2) 2.89 101 What is the measure of A? (1) 51Β° (3) 39Β° (2) 34Β° 9. The y-intercept of the graph of 2x 3y 6 is (2 |
) 2 (3) 3 (1) 6 (4) 56Β° (4) 3 10. When 2a2 5a is subtracted from 5a2 1, the difference is (1) 3a2 5a 1 (2) 3a2 5a 1 (3) 3a2 6a (4) 3a2 5a 1 Part II Answer all questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 11. A straight line has a slope of 3 and contains the point (0, 6). What are the coordinates of the point at which the line intersects the x-axis? 12. Last year the Edwards family spent $6,200 for food. This year, the cost of food for the family was $6,355. What was the percent of increase in the cost of food for the Edwards family? Part III Answer all questions in this part. Each correct answer will receive 3 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 13. Dana paid $3.30 for 2 muffins and a cup of coffee. At the same prices, Damion paid $5.35 for 3 muffins and 2 cups of coffee. What is the cost of a muffin and of a cup of coffee? 14. In trapezoid ABCD,. If AB 54.8 feet, BC'AB and BC'DC DC 37.2 feet, and BC 15.8 feet, find to the nearest degree the measure of A. Cumulative Review 441 Part IV Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 15. Tamara wants to develop her own pictures but does not own photographic equipment. The Community Darkroom makes the use of photographic equipment available for a fee. Membership costs $25 a month and members pay $1.50 an hour for the use of the equipment. Non-members may also use the equipment for $4.00 an hour. a. |
Draw a graph that shows the cost of becoming a member and using the photographic equipment. Let the vertical axis represent cost in dollars and the horizontal axis represent hours of use. b. On the same graph, show the cost of using the photographic equipment for non-members. c. When Tamara became a member of the Community Darkroom in June, she found that her cost for that month would have been the same if she had not been a member. How many hours did Tamara use the facilities of the darkroom in June? 16. ABCD is a rectangle, E is a point on centimeters, and EC 10 centimeters. a. Find the area of nABE. DC, AD 24 centimeters, DE 32 b. Find the perimeter of nABE. CHAPTER 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the Difference of Two Perfect Squares 11-6 Multiplying Binomials 11-7 Factoring Trinomials 11-8 Factoring a Polynomial Completely Chapter Summary Vocabulary Review Exercises Cumulative Review 442 SPECIAL PRODUCTS AND FACTORS The owners of a fruit farm intend to extend their orchards by planting 100 new apple trees.The trees are to be planted in rows, with the same number of trees in each row. In order to decide how the trees are to be planted, the owners will use whole numbers that are factors of 100 to determine the possible arrangements: 20 rows of 5 trees each 1 row of 100 trees 25 rows of 4 trees each 2 rows of 50 trees each 50 rows of 2 trees each 4 rows of 25 trees each 100 rows of 1 tree each 5 rows of 20 trees each 10 rows of 10 trees each From this list of possibilities, the arrangement that best fits the dimensions of the land to be planted can be chosen. If the owner had intended to plant 90 trees, how many possible arrangements would there be? From earliest times, the study of factors and prime numbers has fascinated mathematicians, leading to the discovery of many important principles. In this chapter you will extend your knowledge of the factors of whole numbers, study the products of special binomials, and learn to write polynomials in factored form. 11-1 FACTORS AND FACTORING Factors and Factoring 443 When two numbers are multiplied, the result is called |
their product. The numbers that are multiplied are factors of the product. Since 3(5) 15, the numbers 3 and 5 are factors of 15. Factoring a number is the process of finding those numbers whose product is the given number. Usually, when we factor, we are finding the factors of an integer and we find only those factors that are integers. We call this factoring over the set of integers. Factors of a product can be found by using division. Over the set of integers, if the divisor and the quotient are both integers, then they are factors of the dividend. For example, 35 5 7. Thus, 35 5(7), and 5 and 7 are factors of 35. Every positive integer that is the product of two positive integers is also the product of the opposites of those integers. 21 (3)(7) 21 (3)(7) Every negative integer that is the product of a positive integer and a nega- tive integer is also the product of the opposites of those integers. 21 (3)(7) 21 (3)(7) Usually, when we factor a positive integer, we write only the positive inte- gral factors. Two factors of any number are 1 and the number itself. To find other integral factors, if they exist, we use division, as stated above. We let the number being factored be the dividend, and we divide this number in turn by the whole numbers 2, 3, 4, and so on. If the quotient is an integer, then both the divisor and the quotient are factors of the dividend. For example, use a calculator to find the integral factors of 126. We will use integers as divisors and look for quotients that are integers. Pairs of factors of 126 are listed to the right of the quotients. Quotients 126 1 126 126 2 63 126 3 42 126 4 31.5 126 5 25.2 126 6 21 Pairs of Factors of 126 1 126 2 63 3 42 β β 6 21 Quotients 126 7 18 126 8 15.75 126 9 14 126 10 12.6 126 4 11 5 11.45 126 12 10.5 Pairs of Factors of 126 7 18 β 9 14 β β β When the quotient is smaller than the divisor (here, 10.5 12), we have found all possible positive integral factors. The factors of 126 are 1, 2, 3, 6, 7, 9, 14, 18 |
, 21, 42, 63, and 126. Recall that a prime number is an integer greater than 1 that has no positive integral factors other than itself and 1. The first seven prime numbers are 2, 3, 5, 7, 11, 13, and 17. Integers greater than 1 that are not prime are called composite numbers. 444 Special Products and Factors In general, a positive integer greater than 1 is a prime or can be expressed as the product of prime factors. Although the factors may be written in any order, there is one and only one combination of prime factors whose product is a given composite number. As shown below, a prime factor may occur in the product more than once. 21 3 7 20 2 2 5 or 22 5 To express a positive integer, for example 280, as the product of primes, we start with any pair of positive integers, say 28 and 10, whose product is the given number. Then, we factor these factors and continue to factor the factors until all are primes. Finally, we rearrange these factors in numerical order, as shown at the right. Expressing each of two integers as the product of prime factors makes it possible to discover the greatest integer that is a factor of both of them. We call this factor the greatest common factor (GCF) of these integers. 280 280 28 10 280 2 14 2 5 280 2 2 7 2 5 or 280 23 5 7 Let us find the greatest common factor of 180 and 54. 180 2 2 3 3 5 β β β 54 2 3 3 3 β β 3 3 β Greatest common factor 2 or or or 22 32 5 2 33 2 32 or 18 Only the prime numbers 2 and 3 are factors of both 180 and 54. We see that the greatest number of times that 2 appears as a factor of both 180 and 54 is once; the greatest number of times that 3 appears as a factor of both 180 and 54 is twice. Therefore, the greatest common factor of 180 and 54 is 2 3 3, or 2 32, or 18. To find the greatest common factor of two or more monomials, find the product of the numerical and variable factors that are common to the monomials. For example, let us find the greatest common factor of 24a3b2 and 18a2b. 24a3b2 2 2 2 3 β 18a2b 6a2b Greatest common factor 2 The greatest common factor of 24a3b2 and 18a2b |
is 6a2b. Factors and Factoring 445 When we are expressing an algebraic factor, such as 6a2b, we will agree that: β’ Numerical coefficients need not be factored. (6 need not be written as 2 3.) β’ Powers of variables need not be represented as the product of several equal factors. (a2b need not be written as a a b.) EXAMPLE 1 Solution Write all positive integral factors of 72. Quotients 72 1 72 72 2 36 72 3 24 72 4 18 Pairs of Factors of 72 1 72 2 36 3 24 4 18 Quotients 72 5 14.4 72 6 12 72 4 7 5 10.285714 72 8 9 Pairs of Factors of 72 ββ 6 12 ββ 8 9 It is not necessary to divide by 9, since 9 has just been listed as a factor. Answer The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. EXAMPLE 2 Solution Express 700 as a product of prime factors. 700 2 350 700 2 2 175 700 2 2 5 35 700 2 2 5 5 7 or 22 52 7 Answer 22 52 7 EXAMPLE 3 Find the greatest common factor of the monomials 60r2s4 and 36rs2t. Solution To find the greatest common factor of two monomials, write each as the product of primes and variables to the first power and choose all factors that occur in each product. 60r2s4 36rs2t 2 2 3 3 r s s t Greatest common factor 2 2 3 r s s Answer 12rs2 446 Special Products and Factors EXERCISES Writing About Mathematics 1. Ross said that some pairs of number such as 5 and 12 have no greatest common factor. Do you agree with Ross? Explain why or why not. 2. To find the factors of 200, Chaz tried all of the integers from 2 to 14. Should Chaz have used more numbers? Explain why or why not. Developing Skills In 3β12, tell whether each integer is prime, composite, or neither. 3. 5 8. 36 4. 8 9. 41 5. 13 10. 49 6. 18 11. 57 In 13β22, express each integer as a product of prime numbers. 13. 35 18. 400 14. 18 19. 202 15. 144 20. 129 16. 77 21. 590 7. 73 12 |
. 1 17. 128 22. 316 In 23β34, write all the positive integral factors of each given number. 23. 26 29. 37 24. 50 30. 62 25. 36 31. 253 26. 88 32. 102 27. 100 33. 70 28. 242 34. 169 35. The product of two monomials is 36x3y4. Find the second factor if the first factor is: a. 3x2y3 b. 6x3y2 c. 12xy4 d. 9x3y e. 18x3y2 36. The product of two monomials is 81c9d12e. Find the second factor if the first factor is: a. 27e b. c2d11 c. 3cd3e d. 9c4d4 e. 81c5d9e In 37β42, find, in each case, the greatest common factor of the given integers. 37. 10; 15 40. 18; 24; 36 38. 12; 28 41. 75; 50 39. 14; 35 42. 72; 108 In 43β51, find, in each case, the greatest common factor of the given monomials. 43. 4x; 4y 46. 10x2; 15xy2 49. 14a2b; 13ab 44. 4r; 6r2 47. 36xy2z; β27xy2z2 50. 36xyz; 25xyz 45. 8xy; 6xz 48. 24ab2c3; 18ac2 51. 2ab2c; 3x2yz 11-2 COMMON MONOMIAL FACTORS Common Monomial Factors 447 To factor a polynomial over the set of integers means to express the given polynomial as the product of polynomials whose coefficients are integers. For example, since 2(x y), the polynomial 2x 2y can be written in factored form as 2(x y). The monomial 2 is a factor of each term of the polynomial 2x 2y. Therefore, 2 is called a common monomial factor of the polynomial 2x 2y. 2x 1 2y To factor a polynomial, we look first for the greatest common monomial factor, that is, the greatest monomial that is a factor of each term of the polynomial. For example: 1. Factor 4rs 8st. There are |
many common factors of 4rs and 8st such as 2, 4, 2s, and 4s. The greatest common monomial factor is 4s. We divide 4rs 8st by 4s to obtain the quotient r 2t, which is the second factor. Therefore, the polynomial 4rs 8st 4s(r 2t). 2. Factor 3x 4y. We notice that 1 is the only common factor of 3x and 4y, so the second factor is 3x 4y. We say that 3x 4y is a prime polynomial. A polynomial with integers as coefficients is a prime polynomial if its only factors are 1 and the polynomial itself. Procedure To factor a polynomial whose terms have a common monomial factor: 1. Find the greatest monomial that is a factor of each term of the polynomial. 2. Divide the polynomial by the factor found in step 1.The quotient is the other factor. 3. Express the polynomial as the product of the two factors. We can check by multiplying the factors to obtain the original polynomial. EXAMPLE 1 Write in factored form: 6c3d 12c2d2 3cd. Solution (1) 3cd is the greatest common factor of 6c3d, β12c2d2, and 3cd. (2) To find the other factor, divide 6c3d 12c2d2 3cd by 3cd. 3cd 1 3cd 3cd (6c3d β 12c2d2 3cd) 3cd 3cd 2 12c2d2 6c3d 2c2 4cd 1 Answer 3cd(2c2 4cd 1) 448 Special Products and Factors EXERCISES Writing About Mathematics 1. RamΓ³n said that the factored form of 3a 6a2 15a3 is 3a(2a 5a2). Do you agree with RamΓ³n? Explain why or why not. 2. a. The binomial 12a2 20ab can be written as 4a(3a 5b). Does that mean that for all positive integral values of a and b, the value of 12a2 20ab has at least two factors other than itself and 1? Justify your answer. b. The binomial 3a 5b is a prime polynomial. Does that mean that for all positive inte- gral values of a |
and b, 3a 5b is a prime number? Justify your answer. 3. 2a 2b Developing Skills In 3β29, write each expression in factored form. 4. 3x 3y 7. 4x 8y 10. 18c 27d 13. 6 18c 16. ax 5ab 19. 2x 4x3 22. pr2 2prh 25. 3ab2 6a2b 28. c3 c2 2c 6. xc β xd 9. 12x 18y 12. 7y 7 15. 2x2 5x 18. 10x 15x3 21. pr2 prl 24. 12y2 4y 27. 3x2 6x 30 5. bx by 8. 3m 6n 11. 8x 16 14. y2 3y 17. 3y4 3y2 20. p prt 23. 3a2 9 26. 21r3s2 14r2s 29. 9ab2 6ab 3a Applying Skills 30. The perimeter of a rectangle is represented by 2l 2w. Express the perimeter as the prod- uct of two factors. 31. The lengths of the parallel sides of a trapezoid are represented by a and b and its height by. Express this area as the product of 2ah 1 1 1 2bh h. The area of the trapezoid can be written as two factors. 32. A cylinder is cut from a cube whose edge measures 2s. a. Express the volume of the cube in terms of s. b. If the largest possible cylinder is made, express the volume of the cylinder in terms of s. c. Use the answers to parts a and b to express, in terms of s, the volume of the material cut away to form the cylinder. d. Express the volume of the waste material as the product of two factors. 11-3 THE SQUARE OF A MONOMIAL The Square of a Monomial 449 To square a monomial means to multiply the monomial by itself. For example: (3x)2 (3x)(3x) (3)(3)(x)(x) (3)2(x)2 or 9x2 (5y2)2 (5y2)(5y2) (5)(5)(y2)(y2) (5)2(y2)2 or 25y4 (6b4)2 (6b4)( |
6b4) (6)(6)(b4)(b4) (6)2( b4)2 or 36b8 (4c2d3)2 (4c2d3)(4c2d3) (4)(4)(c2)(c2)(d3)(d3) (4)2(c2)2(d3)2 or 16c4d6 When a monomial is a perfect square, its numerical coefficient is a perfect square and the exponent of each variable is an even number. This statement holds true for each of the results shown above. Procedure To square a monomial: 1. Find the square of the numerical coefficient. 2. Find the square of each literal factor by multiplying its exponent by 2. 3. The square of the monomial is the product of the expressions found in steps 1 and 2. EXAMPLE 1 Square each monomial mentally. a. (4s3)2 2 2 5 ab b. c. (7xy2)2 B A Think (4)2(s3)2 2(a)2(b)2 2 5 A (7)2(x)2(y2)2 B d. (0.3y2)2 (0.3)2(y2)2 EXERCISES Writing About Mathematics Write 16s6 4 25a2b2 49x2y4 0.09y4 1. Explain why the exponent of each variable in the square of a monomial is an even number. 2. When the square of a number is written in scientific notation, is the exponent of 10 always an even number? Explain why or why not and give examples to justify your answer. 450 Special Products and Factors Developing Skills In 3β22, square each monomial. 3. (a2)2 7. (m2n2)2 11. (9ab)2 15. 2 5 7 xy B A 19. (0.8x)2 4. (b3)2 8. (x3y2)2 12. (10x2y2)2 27 8 a2b2 A 20. (0.5y2)2 16. B 2 5. (d5)2 9. (3x2)2 13. (12cd3)2 2 17. x 6 B 21. (0.01xy)2 A 6. (rs)2 10. (5y |
4)2 2 A 14. 3 4 a B 24x2 18. 5 22. (0.06a2b)2 A B 2 Applying Skills In 23β28, each monomial represents the length of a side of a square. Write the monomial that represents the area of the square. 23. 4x 24. 10y 25. 2 3x 26. 1.5x 27. 3x2 28. 4x2y3 11-4 MULTIPLYING THE SUM AND THE DIFFERENCE OF TWO TERMS Recall that when two binomials are multiplied, the product contains four terms. This fact is illustrated in the diagram below and is also shown by using the distributive property. a ac ad + b bc bd (a b)(c d) a(c d) b(c d) ac ad bc bd If two of the four terms of the product are similar, the similar terms can be combined so that the product is a trinomial. For example: c + d (x 2)(x 3) x(x 3) 2(x 3) x2 3x 2x 6 x2 5x 6 There is, however, a special case in which the sum of two terms is multiplied by the difference of the same two terms. In this case, the sum of the two middle terms of the product is 0, and the product is a binomial, as in the following examples: (a 4)(a 4) a(a 4) 4(a 4) a2 4a 4a 16 a2 16 Multiplying the Sum and the Difference of Two Terms 451 (3x2 5y)(3x2 5y) 3x2(3x2 5y) 5y(3x2 5y) 9x4 15x2y 15x2y 25y2 9x4 25y2 These two examples illustrate the following procedure, which enables us to find the product of the sum and difference of the same two terms mentally. Procedure To multiply the sum of two terms by the difference of the same two terms: 1. Square the first term. 2. From this result, subtract the square of the second term. (a b)(a b) a2 b2 EXAMPLE 1 Find each product. a. (y 7)(y 7) b. (3a 4b)(3a 4b) Think (y)2 (7)2 (3a)2 |
(4b)2 Write y2 49 9a2 16b2 EXERCISES Writing About Mathematics 1. Rose said that the product of two binomials is a binomial only when the two binomials are the sum and the difference of the same two terms. Miranda said that that cannot be true because (5a 10)(a 2) 5a2 20, a binomial. Show that Rose is correct by writing the factors in the example that Miranda gave in another way. 2. Ali wrote the product (x 2)2(x 2)2 as (x2 4)2. Do you agree with Ali? Explain why or why not. Developing Skills In 3β17, find each product. 3. (x 8)(x 8) 6. (12 a)(12 a) 9. (8x 3y)(8x 3y) 4. (y 10)(y 10) 7. (c d)(c d) 10. (x2 + 8)(x2 8) 5. (n 9)(n 9) 8. (3x 1)(3x 1) 11. (3 5y3)(3 5y3) 452 Special Products and Factors a 1 1 2 12. 15. (a 5)(a 5)(a2 25) B A A B a 2 1 2 13. (r 0.5)(r 0.5) 16. (x 3)(x 3)(x2 9) 14. (0.3 m)(0.3 m) 17. (a b)(a b)(a2 b2) Applying Skills In 18β21, express the area of each rectangle whose length l and width w are given. 18. l x 7, w x 7 19. l 2x 3, w 2x 3 20. l c d, w c d 21. l 2a 3b, w 2a 3b In 22β25, find each product mentally by thinking of the factors as the sum and difference of the same two numbers. 22. (27)(33) (30 3)(30 3) 23. (52)(48) (50 2)(50 2) 24. (65)(75) 25. (19)(21) 11-5 FACTORING THE DIFFERENCE OF TWO PERFECT SQUARES An expression of the form a2 b2 is called a difference of two perfect squares. Factoring the difference of two perfect squares is the reverse of |
multiplying the sum of two terms by the difference of the same two terms. Since the product (a b)(a b) is a2 b2, the factors of a2 b2 are (a b) and (a b). Therefore: a2 b2 (a b)(a b) Procedure To factor a binomial that is a difference of two perfect squares: 1. Express each of its terms as the square of a monomial. 2. Apply the rule a2 β b2 (a b)(a β b). Remember that a monomial is a square if and only if its numerical coeffi- cient is a square and the exponent of each of its variables is an even number. EXAMPLE 1 Factor each polynomial. a. r2 1 25x2 2 1 49y2 b. c. 0.04 c6d4 Think (r)2 (1)2 Write (r 1)(r 1) 2 1 7y (5x)2 2 A (0.2)2 (c3d 2)2 B 5x 1 1 7y A (0.2 c3d 2)(0.2 c3d 2) 5x 2 1 7y B A B Factoring the Difference of Two Perfect Squares 453 EXAMPLE 2 Express x2 β 100 as the product of two binomials. Solution Since x2 100 is a difference of two squares, the square of x and the square of 10, the factors of x2 100 are (x 10) and (x 10). Answer (x 10)(x 10) EXERCISES Writing About Mathematics 1. If 391 can be written as 400 9, find two factors of 391 without using a calculator or pencil and paper. Explain how you found your answer. 2. Does 5a2 45 have two binomial factors? Explain your answer. Developing Skills In 3β18, factor each binomial. 3. a2 4 7. 16a2 b2 11. 25 s4 15. 0.04 49r2 4. c2 100 8. 25m2 n2 12. 100x2 81y2 16. 0.16y2 9 5. 9 x2 9. d 2 4c2 w2 2 1 13. 64 17. 0.81 y2 6. 144 c2 10. r4 9 14. x2 0.64 18. 81m4 49 Applying Skills In 19β23, each |
given polynomial represents the area of a rectangle. Express the area as the product of two binomials. 19. x2 4 23. 4x2 y2 22. t4 64 21. t2 49 20. y2 9 In 24β26, express the area of each shaded region as: a. the difference of the areas shown, and b. the product of two binomials. 24. c d c d d c 25. 2x 26. x d c 2x x y y y y y y 2x 2x 454 Special Products and Factors In 27 and 28, express the area of each shaded region as the product of two binomials. 27. 5a 28. 2b 2b 2b 2b 5a 3x 2y 2y 2y 2y 2y 2y 2y 2y 3x 11-6 MULTIPLYING BINOMIALS We have used the βFOILβ method from Section 5-4 to multiply two binomials of the form ax b and cx d. The pattern of the multiplication in the following example shows us how to find the product of two binomials mentally. β€ (2x 3)(4x 5) 2x(4x) 2x(5) 3(4x) 3(5) β€ β€ β€ 8x2 10x 12x 15 8x2 2x 15 Examine the trinomial 8x2 2x 15 and note the following: 1. The first term of the trinomial is the product of the first terms of the binomials: 2. The middle term of the trinomial is the product of the outer terms plus the product of the inner terms of the binomials: 3. The last term of the trinomial is the product of the last terms of the binomials: 8x2 |________| (2x 3)(4x 5) β 2x(4x) 8x2 12x Think |____| (2x 3)(4x 5) β (12x) (10x) 2x |______________| 10x (2x 3)(4x 5) β (3)(5) 15 |_________| 15 Multiplying Binomials 455 Procedure To find the product of two binomials (ax b) and (cx d) where a, b, c, and d have numerical values |
: 1. Multiply the first terms of the binomials. 2. Multiply the first term of each binomial by the last term of the other bino- mial (the outer and inner terms), and add these products. 3. Multiply the last terms of the binomials. 4. Write the results of steps 1, 2, and 3 as a sum. Write the product (x 5)(x 7) as a trinomial. β€ (x 5)(x 7) β€ β€ β€ 1. (x)(x) x2 2. (x)(7) (5)(x) (7x) (5x) β12x 3. (5)(7) 35 4. x2 12x 35 Answer Write the product (3y 8)(4y 3) as a trinomial. β€ β€ (3y 8)(4y 3) β€ β€ 1. (3y)(4y) 12y2 2. (3y)(3) (8)(4y) 9y (32y) β23y 3. (8)(3) β24 4. 12y2 23y 24 Answer EXAMPLE 1 Solution EXAMPLE 2 Solution EXAMPLE 3 Write the product (z 4)2 as a trinomial. Solution (z 4)2 (z 4)(z 4) z2 4z 4z 16 z2 8z 16 Answer A trinomial, such as z2 8z 16, that is the square of a binomial is called a perfect square trinomial. Note that the first and last terms are the squares of the terms of the binomial, and the middle term is twice the product of the terms of the binomial. 456 Special Products and Factors EXERCISES Writing About Mathematics 1. What is a shorter procedure for finding the product of binomials of the form (ax b) and (ax b)? 2. Which part of the procedure for multiplying two binomials of the form (ax b) and (cx d) does not apply to the product of binomials such as (3x 5) and (2y 7)? Explain your answer. Developing Skills In 3β29, perform each indicated operation mentally. 4. (6 d)(3 d) 7. (x 7)(x 2) 10. (3x 2)(x 5) 13 |
. (a 4)2 16. (7x 3)(2x 1) 19. (2x 5)2 22. (2t 3)(5t 1) 25. (2c 3d)(5c 2d) 28. (6t 1)(4t z) 3. (x 5)(x 3) 6. (8 c)(3 c) 9. (5 t)(9 t) 12. (y 8)2 15. (3x 2)2 18. (3x 4)2 21. (5y 4)(5y 4) 24. (5x 7y)(3x 4y) 27. (a b)(2a 3) Applying Skills 5. (x 10)(x 5) 8. (n 20)(n 3) 11. (c 5)(3c 1) 14. (2x 1)2 17. (2y 3)(3y 2) 20. (3t 2)(4t 7) 23. (2a b)(2a 3b) 26. (a b)2 29. (9y w)(9y 3w) 30. Write the trinomial that represents the area of a rectangle whose length and width are: a. (x 5) and (x 4) b. (2x 3) and (x 1) 31. Write the perfect square trinomial that represents the area of a square whose sides are: a. (x 6) b. (x 2) c. (2x 1) d. (3x 2) 32. The length of a garden is 3 feet longer than twice the width. a. If the width of the garden is represented by x, represent the length of the garden in terms of x. b. If the width of the garden is increased by 5 feet, represent the new width in terms of x. c. If the length of the enlarged garden is also increased so that it is 3 feet longer than twice the new width, represent the length of the enlarged garden in terms of x. d. Express the area of the enlarged garden in terms of x. 11-7 FACTORING TRINOMIALS Factoring Trinomials 457 (x 1 3) We have learned that (x 3)(x 5) x2 8x 15. Therefore, factors of x2 8x 15 are and (x + 5). Factoring a trinomial of the form ax2 + bx c is |
the reverse of multiplying binomials of the form (dx e) and (fx g). When we factor a trinomial of this form, we use combinations of factors of the first and last terms. We list the possible pairs of factors, then test the pairs, one by one, until we find the pair that gives the correct middle term. For example, let us factor x2 7x 10: 1. The product of the first terms of the binomials must be x2. Therefore, for each first term, we use x. We write: x2 7x 10 (x )(x ) 2. Since the product of the last terms of the binomials must be +10, these last terms must be either both positive or both negative. The pairs of integers whose products is +10 are: (1)(10) (5)(2) (1)(10) (2)(5) 3. From the products obtained in steps 1 and 2, we see that the possible pairs of factors are: (x 10)(x 1) (x 10)(x 1) (x 5)(x 2) (x 5)(x 2) 4. Now, we test each pair of factors: (x 10)(x 1) is not correct because the middle term is 11x, not 7x. (x 5)(x 2) is correct because the middle term is 7x. 10x |____| (x 10)(x 1) β 10x 1x 11x |_____________| β 1x 5x |___| (x 5)(x 2) β 2x 5x 7x |____________| β 2x Neither of the remaining pairs of factors is correct because each would result in a negative middle term. 5. The factors of x2 7x 10 are (x 5) and (x 2). Observe that in this trinomial, the first term, x2, is positive and must have factors with the same signs. We usually choose positive factors of the first terms. The last term, +10, is positive and must have factors with the same signs. Since the middle term of the trinomial is also positive, the last terms of both binomial factors must be positive (5 and 2). In this example, if we had chosen x times x as the factors of x2, the factors of x2 7x 10 would have been written as (x 5) and (x 2). Every trinomial that has two |
binomial factors also has the opposites of these binomials as factors. Usually, however, we write only the pair of factors whose first terms have positive coefficients as the factors of a trinomial. 458 Special Products and Factors Procedure To factor a trinomial of the form ax2 bx c, find two binomials that have the following characteristics: 1. The product of the first terms of the binomials is equal to the first term of the trinomial (ax2). 2. The product of the last terms of the binomials is equal to the last term of the trinomial (c). 3. When the first term of each binomial is multiplied by the last term of the other binomial and the sum of these products is found, the result is equal to the middle term of the trinomial. EXAMPLE 1 Factor y2 8y 12. Solution (1) The product of the first terms of the binomials must be y2. Therefore, for each first term, we use y. We write: y2 8y 12 (y )(y ) (2) Since the product of the last terms of the binomials must be +12, these last terms must be either both positive or both negative. The pairs of integers whose product is +12 are: (1)(12) (1)(12) (2)(6) (2)(6) (3)(4) (3)(4) (3) The possible factors are: (y l)(y 12) (y l)(y 12) (y 2)(y 6) (y 2)(y 6) (y 3)(y 4) (y 3)(y 4) (4) When we find the middle term in each of the trinomial products, we see that only for the factors (y 6)(y 2) is the middle term 8y: 6y |____| (y 6)(y 2) β 2y 6y 8y |____________| 2y Answer y2 8y 12 (y 6)(y 2) When the first and last terms are both positive (y2 and 12 in this example) and the middle term of the trinomial is negative, the last terms of both binomial factors must be negative (6 and β2 in this example). Factoring Trinomials 459 EXAMPLE 2 Factor: a. c2 5c 6 b. c2 5c 6 Solution a. (1 |
) The product of the first terms of the binomials must be c2. Therefore, for each first term, we use c. We write: c2 5c 6 (c )(c ) (2) Since the product of the last terms of the binomials must be β6, one of these last terms must be positive and the other negative. The pairs of integers whose product is β6 are: (1)(6) (1)(6) (3)(2) (3)(2) (3) The possible factors are: (c 1)(c 6) (c 1)(c 6) (c 3)(c 2) (c 3)(c 2) (4) When we find the middle term in each of the trinomial products, we see that only for the factors (c 1)(c 6) is the middle term 5c: 1c |____| (c 1)(c 6) β 6c 1c 5c |____________| 6c b. (1) As in the solution to part a, we write: c2 5c 6 (c )(c ) (2) Since the product of the last terms of the binomials must be 6, these last terms must be either both positive or both negative. The pairs of integers whose product is 6 are: (1)(6) (1)(6) (3)(2) (3)(2) (3) The possible factors are: (c 1)(c 6) (c 1)(c 6) (c 3)(c 2) (c 3)(c 2) (4) When we find the middle term in each of the trinomial products, we see that only for the factors (c 3)(c 2) is the middle term 5c. 3c |____| (c 3)(c 2) β 2c 3c 5c |____________| 2c Answer a. c2 5c 6 (c 1)(c 6) b. c2 5c 6 (c 3)(c 2) 460 Special Products and Factors EXAMPLE 3 Factor 2x2 7x 15. Solution (1) Since the product of the first terms of the binomials must be 2x2, we use 2x and x as the first terms. We write: 2x2 7x 15 (2x )(x ) (2) Since the product of the last terms of the binomials must be 15, one of these |
last terms must be positive and the other negative. The pairs of integers whose product is β15 are: (1)(15) (1)(15) (3)(5) (3)(5) (3) These four pairs of integers will form eight pairs of binomial factors since there are two ways in which the first terms can be arranged. Note how (2x 1)(x 15) is not the same product as (2x 15)(x 1). The possible pairs of factors are: (2x 1)(x 15) (2x 15)(x 1) (2x 1)(x 15) (2x 15)(x 1) (2x 3)(x 5) (2x 5)(x 3) (2x 3)(x 5) (2x 5)(x 3) (4) When we find the middle term in each of the trinomial products, we see that only for the factors (2x 3)(x 5) is the middle term 7x: 3x |____| (2x 3)(x 5) β 3x 10x 7x |_____________| 10x Answer 2x2 7x 15 (2x 3)(x 5) In factoring a trinomial of the form ax2 bx c, when a is a positive inte- ger (a 0): 1. The coefficients of the first terms of the binomial factors are usually writ- ten as positive integers. 2. If the last term, c, is positive, the last terms of the binomial factors must be either both positive (when the middle term, b, is positive), or both negative (when the middle term, b, is negative). 3. If the last term, c, is negative, one of the last terms of the binomial factors must be positive and the other negative. Factoring a Polynomial Completely 461 EXERCISES Writing About Mathematics 1. Alicia said that the factors of x2 bx c are (x d)(x e) if c de and b d e. Do you agree with Alicia? Explain why or why not. 2. Explain why there is no positive integral value of c for which x2 x c has two binomial factors. Developing Skills In 3β33, factor each trinomial. 3. a2 3a 2 7. y2 6y 8 11. y2 2y 8 15. x2 11x 24 19. x2 10x 24 23. |
c2 2c 35 27. 3x2 10x 8 31. 10a2 9a 2 4. c2 6c 5 8. y2 6y 8 12. y2 2y 8 16. a2 11a 18 20. x2 x 2 24. x2 7x 18 28. 16x2 8x 1 32. 3a2 7ab 2b2 5. x2 8x 7 9. y2 9y 8 13. y2 7y β 8 17. z2 10z 25 21. x2 6x 7 25. z2 9z 36 29. 2x2 x 3 33. 4x2 5xy 6y2 6. x2 11x 10 10. y2 9y 8 14. y2 7y 8 18. x2 5x 6 22. y2 4y 5 26. 2x2 5x 2 30. 4x2 12x 5 Applying Skills In 34β36, each trinomial represents the area of a rectangle. In each case, find two binomials that could be expressions for the dimensions of the rectangle. 34. x2 10x 9 35. x2 9x 20 36. 3x2 14x 15 In 37β39, each trinomial represents the area of a square. In each case, find a binomial that could be an expression for the measure of each side of the square. 37. x2 10x 25 38. 81x2 18x 1 39. 4x2 + 12x 9 11-8 FACTORING A POLYNOMIAL COMPLETELY Some polynomials, such as x2 + 4 and x2 + x 1, cannot be factored into other polynomials with integral coefficients. We say that these polynomials are prime over the set of integers. Factoring a polynomial completely means finding the prime factors of the polynomial over a designated set of numbers. In this book, we will consider a polynomial factored when it is written as a product of monomials or prime polynomials over the set of integers. 462 Special Products and Factors Procedure To factor a polynomial completely: 1. Look for the greatest common factor. If it is greater than 1, factor the given polynomial into a monomial times a polynomial. 2. Examine the polynomial factor or the given polynomial if it has no common factor (the greatest common |
factor is 1).Then: β’ Factor any trinomial into the product of two binomials if possible. β’ Factor any binomials that are the difference of two perfect squares as such. 3. Write the given polynomial as the product of all of the factors. Make certain that all factors except the monomial factor are prime polynomials. EXAMPLE 1 Factor by2 4b. Solution How to Proceed (1) Find the greatest common factor of the terms: by2 4b b(y2 4) (2) Factor the difference of by2 4b b(y 2)(y 2) Answer two squares: EXAMPLE 2 Factor 3x2 β 6x 24. Solution How to Proceed (1) Find the greatest common 3x2 β 6x 24 3(x2 β 2x 8) factor of the terms: (2) Factor the trinomial: 3x2 β 6x 24 3(x 4)(x 2) Answer EXAMPLE 3 Factor 4d2 6d 2. Solution How to Proceed (1) Find the greatest common 4d2 6d 2 2(2d2 3d 1) factor of the terms: (2) Factor the trinomial: 4d2 6d 2 2(2d 1)(d 1) Answer Factoring a Polynomial Completely 463 EXAMPLE 4 Factor x4 16. Solution How to Proceed (1) Find the greatest common The greatest common factor is 1. factor of the terms: (2) Factor the binomial as the difference of two squares: x4 16 (x2 4)(x2 4) (3) Factor the difference of x4 16 (x2 4)(x 2)(x 2) Answer two squares: EXERCISES Writing About Mathematics 1. Greta said that since 4a2 a2b2 is the difference of two squares, the factors are. Has Greta factored 4a2 a2b2 into prime polynomial factors? Explain (2a 1 ab)(2a 2 ab) why or why not. 2. Raul said that the factors of x3 1 are (x 1)(x 1)(x 1). Do you agree with Raul? Explain why or why not. Developing Skills In 3β29, factor each polynomial completely. 4. 4x2 4 7. 2x2 32 10. 63c2 7 13. 4a2 36 16. pc2 |
pd2 19. x3 7x2 10x 22. 2ax2 2ax 12a 25. y4 13y2 36 28. 16x2 16x 4 3. 2a2 2b2 6. st2 9s 9. 18m2 8 12. z3 z 15. y4 81 18. 4r2 4r 48 21. d3 8d2 16d 24. a4 10a2 9 27. 2a2b 7ab 3b 5. ax2 ay2 8. 3x2 27y2 11. x3 4x 14. x4 1 17. 3x2 6x 3 20. 4x2 6x 4 23. 16x2 x2y4 26. 5x4 10x2 5 29. 25x2 100xy 100y2 464 Special Products and Factors Applying Skills 30. The volume of a rectangular solid is represented by 12a3 5a2b 2ab2. Find the algebraic expressions that could represent the dimensions of the solid. 31. A rectangular solid has a square base. The volume of the solid is represented by 3m2 1 12m 1 12. a. Find the algebraic expression that could represent the height of the solid. b. Find the algebraic expression that could represent the length of each side of the base. 32. A rectangular solid has a square base. The volume of the solid is represented by 10a3 1 20a2 1 10a. a. Find the algebraic expression that could represent the height of the solid. b. Find the algebraic expression that could represent the length of each side of the base CHAPTER SUMMARY Factoring a number or a polynomial is the process of finding those numbers or polynomials whose product is the given number or polynomial. A perfect square trinomial is the square of a binomial, whereas the difference of two perfect squares is the product of binomials that are the sum and difference of the same two. A prime polynomial, like a prime number, has only two factors, 1 and itself. To factor a polynomial completely: 1. Factor out the greatest common monomial factor if it is greater than 1. 2. Write any factor of the form a2 β b2 as (a b)(a β b). 3. Write any factor of the form ax2 + bx c as the product of two binomial factors if possible. 4. |
Write the given polynomial as the product of these factors. VOCABULARY 11-1 Product β’ Factoring a number β’ Factoring over the set of integers β’ Greatest common factor 11-2 Factoring a polynomial β’ Common monomial factor β’ Greatest common monomial factor β’ Prime polynomial 11-3 Square of a monomial 11-5 Difference of two perfect squares 11-6 Perfect square trinomial 11-8 Prime over the set of integers β’ Factoring a polynomial completely REVIEW EXERCISES Review Exercises 465 1. Leroy said that 4x2 16x 12 (2x 2)(2x 6) 2(x 1)(x 3). Do you agree with Leroy? Explain why or why not. 2. Express 250 as a product of prime numbers. 3. What is the greatest common factor of 8ax and 4ay? 4. What is the greatest common factor of 16a3bc2 and 24a2bc4? In 5β8, square each monomial. 5. (3g3)2 6. (4x4)2 7. (0.2c2y)2 In 9β14, find each product. 8. A 1 2 a3b5 2 B 9. (x 5)(x 9) 12. (3d 1)(d 2) 10. (y 8)(y 6) 13. (2w 1)2 11. (ab 4)(ab 4) 14. (2x 3c)(x 4c) In 15β29, in each case factor completely. 15. 6x 27b 18. x2 16h2 21. 64b2 9 24. a2 7a 30 27. 2x2 12bx 32b2 30. Express the product (k 15)(k 15) as a binomial. 31. Express 4ez2(4e z) as a binomial. 32. Factor completely: 60a2 37a 6. 16. 3y2 + 10y 19. x2 4x 5 22. 121 k2 25. x2 16x 60 28. x4 1 17. m2 81 20. y2 9y 14 23. x2 8x 16 26. 16y2 16 29. 3x3 6x2 24x 33. Which of the following polynomials has a factor that is a perfect square trinomial and a factor that is a perfect square? |
(1) a2y 10ay 25y (2) 2ax2 2ax 12a (3) 18m2 24m 8 (4) c2z2 18cz2 81z2 34. Of the four polynomials given below, explain how each is different from the others. x2 9 x3 5x2 6x 35. If the length and width of a rectangle are represented by 2x 3 and 3x 2, respectively, express the area of the rectangle as a trinomial. x2 2x 1 x2 2x 1 36. Find the trinomial that represents the area of a square if the measure of a side is 8m 1 1. 466 Special Products and Factors 37. If 9x2 30x 25 represents the area of a square, find the binomial that represents the length of a side of the square. 38. A group of people wants to form several committees, each of which will have the same number of persons. Everyone in the group is to serve on one and only one committee. When the group tries to make 2, 3, 4, 5, or 6 committees, there is always one extra person. However, they are able to make more than 6 but fewer than 12 committees of equal size. a. What is the smallest possible number of persons in the group? b. Using the group size found in a, how many persons are on each com- mittee that is formed? Exploration a. Explain how the expression (a b)(a b) a2 b2 can be used to find the product of two consecutive even or two consecutive odd integers. b. The following diagrams illustrate the formula: 1 3 5... (2n 3) (2n 1) n2 1 + 3 = 4 = 22 1 + 3 + 5 = 9 = 32 1 + 3 + 5 + 7 = 16 = 42 (1) Explain how 3, 5, and 7 are the difference of two squares using the diagrams. (2) Use the formula to explain how any odd number, 2n 1, can be written as the difference of squares. [Hint: 1 3 5... (2n 3) (n 1)2] CUMULATIVE REVIEW CHAPTERS 1β11 Part I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. 1. If apples cost $3.75 for 3 pounds, what is the cost |
, at the same rate, of 7 pounds of apples? (1) $7.25 (2) $8.25 (3) $8.50 (4) $8.75 Cumulative Review 467 (1) a2 4a 4 2. When 3a2 4 is subtracted from 2a2 4a, the difference is (2) a2 4a 4 (3) a2 8a 3. The solution of the equation x 3(2x 4) 7x is (4) a2 a (1) 6 (2) 6 (3) 1 (4) 1 4. The area of a circle is 16p square centimeters. The circumference of the circle is (1) 16p centimeters (2) 8p centimeters (3) 4p centimeters (4) 2p centimeters 5. The x-intercept of the line whose equation is 2x y 5 is (1) 5 (2) 5 6. The factors of x2 3x 10 are (1) (x 5)(x 2) (2) (x 5)(x 2) (3) 2 5 (4) 5 2 (3) (x 5)(x 2) (4) (x 5)(x 2) 7. The fraction (1) 5 10β3 2.5 3 1025 5.0 3 1028 is equal to (2) 5 10β2 (3) 5 102 (4) 5 103 8. When factored completely, 5a3 45a is equal to (1) 5a(a 3)2 (2) 5(a2 3)(a 3) (3) a(5a 9)(a 5) (4) 5a(a 3)(a 3) 9. When a 3, a2 5a is equal to (1) 6 (2) 6 (3) 24 (4) 24 10. The graph of the equation 2x y 7 is parallel to the graph of (1) y 2x 3 (2) y β2x 5 (3) x 2y 4 (4) y 2x 2 Part II Answer all questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 11. The measures of the sides of a right triangle are 32, 60, and |
68. Find, to the nearest degree, the measure of the smallest angle of the triangle. 12. Solve and check: x x 1 8 5 2 3. 468 Special Products and Factors Part III Answer all questions in this part. Each correct answer will receive 3 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 13. The width of Mattieβs rectangular garden is 10 feet less than the length. She bought 25 yards of fencing and used all but 7 feet to enclose the garden. a. Write an equation or a system of equations that can be used to find the dimensions, in feet, of the garden. b. What are the dimensions, in feet, of the garden? 14. The population of a small town at the beginning of the year was 7,000. Records show that during the year there were 5 births, 7 deaths, 28 new people moved into town, and 12 residents moved out. What was the percent of increase or decrease in the town population? Part IV Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 15. Admission to a museum is $5.00 for adults and $3.00 for children. In one day $2,400 was collected for 620 paid admissions. How many adults and how many children paid to visit the museum that day? 16. a. Write an equation of a line whose slope is β2 and whose y-intercept is 5. b. Sketch the graph of the equation written in a. c. Are (1, 3) the coordinates of a point on the graph of the equation written in a? d. If (k, 1) are the coordinates of a point on the graph of the equation written in a, what is the value of k? OPERATIONS WITH RADICALS Whenever a satellite is sent into space, or astronauts are sent to the moon, technicians at earthbound space centers monitor activities closely.They continually make small corrections to help the spacecraft stay on course. The distance from the earth to the moon varies, from 221,460 miles to 252,700 miles, and both the earth and the moon are constantly rotating in space |
.A tiny error can send the craft thousands of miles off course.Why do such errors occur? Space centers rely heavily on sophisticated computers, but computers and calculators alike work with approximations of numbers, not necessarily with exact values. We have learned that irrational numbers, such as 2 and 5 ", are shown on a calculator as decimal " approximations of their true values.All irrational numbers, which include radicals, are nonrepeating decimals that never end. How can we work with them? In this chapter, we will learn techniques to compute with radicals to find exact answers. We will also look at methods for working with radicals on a calculator to understand how to minimize errors when using these devices. CHAPTER 12 CHAPTER TABLE OF CONTENTS 12-1 Radicals and the Rational Numbers 12-2 Radicals and the Irrational Numbers 12-3 Finding the Principal Square Root of a Monomial 12-4 Simplifying a Square-Root Radical 12-5 Addition and Subtraction of Radicals 12-6 Multiplication of Square-Root Radicals 12-7 Division of Square-Root Radicals Chapter Summary Vocabulary Review Exercises Cumulative Review 469 470 Operations With Radicals 12-1 RADICALS AND THE RATIONAL NUMBERS Squares and Square Roots Recall from Section 1-3 that to square a number means to multiply the number by itself. To square 8, we write: 82 8 8 64 On a calculator: ENTER: 8 x2 ENTER DISPLAY: 8 2 6 4 To find the square root of a number means to find the value that, when multiplied by itself, is equal to the given number. To express the square root of 64, we write: 64 5 8 " On a calculator: ENTER: 2nd Β― 64 ENTER DISPLAY: β ( 6 4 8 The symbol " sign is the radicand. For example, in 64,β the radicand is 64. " is called the radical sign, and the quantity under the radical, which we read as βthe square root of 64 A radical, which is any term containing both a radical sign and a radicand, is a root of a quantity. For example, 64 is a radical. In general, the square root of b is x (written as " and x2 b. b 5 x ) if and only if x 0 " Some radicals, such as 2 " 3, are irrational numbers. We begin this study of radicals by examining " 9, |
are rational numbers; others, such as " and radicals that are rational numbers. 4 " and Radicals and the Rational Numbers 471 Perfect Squares Any number that is the square of a rational number is called a perfect square. For example, 3 3 9 0 0 0 1.4 1.4 1.96 2 7 3 2 7 5 4 49 Therefore, perfect squares include 9, 0, 1.96, and Then, by applying the inverse operation, we know that: 4 49. The square root of every perfect square is a rational number. 49 5 2 4 7 1.4 3 0 1.96 0 " " # 9 " Radicals That Are Square Roots Certain generalizations can be made for all radicals that are square roots, whether they are rational numbers or irrational numbers: 1. Since the square root of 36 is a number whose square is 36, we can write the statement also true that 2 36 B (6)2 5 A " " 36 5 6. " 5 36. We notice that In general, for every nonnegative real number n: 36 A " 2 B 5 (6)2 5 36. It is n A " B 2 5 n and n2 5 n. " 2. Since (6)(6) 36 and (6)(6) 36, both 6 and 6 are square roots of 36. This example illustrates the following statement: Every positive number has two square roots: one root is a positive number called the principal square root, and the other root is a negative number. These two roots have the same absolute value. To indicate the positive or principal square root only, place a radical sign over the number: 25 5 5 and 0.49 5 0.7 " " To indicate the negative square root only, place a negative sign before the radical: 2 25 5 25 and 2 0.49 5 20.7 To indicate both square roots, place both a positive and a negative sign before the radical: 6 25 5 65 " and 6 0.49 5 60.7 " " " 472 Operations With Radicals 3. Every real number, when squared, is either positive or 0. Therefore: The square root of a negative number does not exist in the set of real numbers. For example, real number that, when multiplied by itself, equals 25. 225 " does not exist in the set of real numbers because there is no Calculators and Square Roots A calculator will return only the principal square root of a positive number. To display the negative square root of a |
number, the negative sign must be entered before the radical. ENTER: (-) 2nd Β― 25 ENTER DISPLAY: - β ( 2 5 - 5 The calculator will display an error message if it is set in βrealβ mode and the square root of a negative number is entered. ENTER: 2nd Β― (-) 25 ENTER DISPLAY Cube Roots and Other Roots A cube root of a number is one of the three equal factors of the number. For example, 2 is a cube root of 8 because 2 2 2 8, or 23 8. A cube root of 8 is written as 3 8. " Finding a cube root of a number is the inverse operation of cubing a ) if and only number. In general, the cube root of b is x (written as if x3 b. 3 b 5 x " 3 28 " then 225 We have said that does not exist in the set of real numbers. However, does exist in the set of real numbers. Since (2)3 (2)(2)(2) 8, 3 28 5 22 " ". Radicals and the Rational Numbers 473 In the set of real numbers, every number has one cube root. The cube root of a positive number is positive, and the cube root of a negative number is negative. In the expression, the integer n that indicates the root to be taken, is called the index of the radical. Here are two examples: n b " β’ In β’ In 3 8 " 4 16 ", read as βthe cube root of 8,β the index is 3., read as βthe fourth root of 16,β the index is 4. Since 24 16, 2 is one of the four equal factors of 16, and 4 16 5 2. " When no index appears, the index is understood to be 2. Thus, 25 5 2 25 5 5 ". " When the index of the root is even and the radicand is positive, the value is a real number. That real number is positive if a plus sign or no sign precedes the radical, and negative if a minus sign precedes the radical. When the index of the root is even and the radicand is negative, the root has no real number value. When the index of the root is odd and the radicand either positive or negative, the value is a real number. That real number is positive if the radicand is positive and negative if the radicand is negative. Calculators |
In 3β22, express each radical as a rational number with the appropriate signs. 16 3. 8. " 2 " 625 4. 9. 64 2 " 1 4 $ 6 5. 6 2 10. " 100 9 16 $ 13. 2 1.44 14. 0.09 15. 6 0.0004 18. " 5 32 " " 3 28 " In 23β32, evaluate each radical by using a calculator. " 3 2125 " 19. 20. 2 6. 6 169 11. 16. 21. " 6 25 81 $ 3 1 " 4 0.1296 " 7. 12. 17. 22. 400 " 0.64 " 4 81 " 2 36 4 $ 2 23. 24. 10.24 " 5 21,024 " 46.24 " 3 21,000 " In 33β38, find the value of each expression in simplest form. 32.49 3 2.197 " 2 25. 28. 29. 30. " 2 26. 31. 3 23,375 " 3 20.125 " 27. 32. 4 4,096 " 6 5.76 " 33. 36. 8 A " $A 2 2 B 34. 37. B 9 3 $A 2 1 2 B 97 A " B A " 35. 38. B 2 0.7 A " B (29)2 1 " 83 A " 2 B 97 476 Operations With Radicals In 39β47, replace each? with,, or to make a true statement. 39. 42 $ 45. m? m " 2 40. 1? 12, 0 m 1 43. 4 25? 46. m? 4 25 $ " m, m 1 41. 3 2? 44. 1? 2 3 2 A B 1 " 47. m? m ", m 1 In 48β55, solve each equation for the variable when the replacement set is the set of real numbers. 48. x2 4 52. y2 30 6 4 81 49. y2 53. 2x2 50 50. x2 0.49 54. 3y2 27 0 51. x2 16 0 55. x3 8 Applying Skills In 56β59, in each case, find the length of the hypotenuse of a right triangle when the lengths of the legs have the given values. 56. 6 inches and 8 inches 58. 15 meters and 20 meters 57. 5 centimeters and 12 centimeters 59. 15 feet and 36 feet In 60β63, find, |
in each case: a. the length of each side of a square that has the given area b. the perimeter of the square. 60. 36 square feet 61. 196 square yards 62. 121 square centimeters 64. Express in terms of x, (x 0), the perimeter of a square whose area is represented by x2. 63. 225 square meters 65. Write each of the integers from 101 to 110 as the sum of the smallest possible number of perfect squares. (For example, 99 72 72 12.) Use positive integers only. 12-2 RADICALS AND THE IRRATIONAL NUMBERS We have learned that type of number is examine n is a rational number when n is a perfect square. What n" when n is not a perfect square? As an example, let us using a calculator. " 5 " 2nd ENTER: DISPLAY: Β― 5 ENTER β ( imation of Can we state that 5 " 2.236067977? Or is 2.236067977 a rational approx? To answer this question, we will find the square of 2.236067977. 5 " Radicals and the Irrational Numbers 477 ENTER: 2.236067977 x2 ENTER DISPLAY We know that if, then x2 b. The calculator displays shown above b 5 x 2.236067977 because (2.236067977)2 5. The rational 5" demonstrate that number 2.236067977 is an approximate value for " Recall that a number such as is called an irrational number. Irrational, where a and b are integers and numbers cannot be expressed in the form b 0. Furthermore, irrational numbers cannot be expressed as terminating or repeating decimals. The example above illustrates the truth of the following statement: " a b 5 5. " If n is any positive number that is not a perfect square, then tional number. is an irra- n " Radicals and Estimation 5 " represents the exact value of the irrational number whose The radical is a rational approximation of the irrasquare is 5. The calculator display for tional number. It is a number that is close to, but not equal to. There are other values correctly rounded from the calculator display that are also approximations of 5 " 5 " 5 : " 2.236067977 2.236068 2.23607 2.2361 2.236 2.24 calculator display, to nine decimal places rounded to six decimal places ( |
nearest millionth) rounded to five decimal places (nearest hundred-thousandth) rounded to four decimal places (nearest ten-thousandth) rounded to three decimal places (nearest thousandth) rounded to two decimal places (nearest hundredth) is greater Each rational approximation of than 2 but less than 3. This fact can be further demonstrated by placing the square 5, which is 5, between the squares of two consecutive integers, one less than of " 5 and one greater than 5, and then finding the square root of each number. 5, as seen above, indicates that 5 " " Since then or " 4 5 9. " " 478 Operations With Radicals In the same way, to get a quick estimate of any square-root radical, we place its square between the squares of two consecutive integers. Then we take the square root of each term to show between which two consecutive integers the 73 : radical lies. For example, to estimate Since 64 73 81, 73 81, then 73, 9". or 64 8, " " " " Basic Rules for Radicals That Are Irrational Numbers There are general rules to follow when working with radicals, especially those that are irrational numbers: 1. If the degree of accuracy is not specified in a question, it is best to give the exact answer in radical form. In other words, if the answer involves a radical, leave the answer in radical form. For example, the sum of 2 and 5 " 2. If the degree of accuracy is not specified in a question and a rational 5, an exact value. is written as 2 1 " approximation is to be given, the approximation should be correct to two or more decimal places. For example, an exact answer is 5. By using a calculator, a student " is approximately 2 2.236067977 4.236067977. discovers that Acceptable answers would include the calculator display and correctly rounded approximations of the display to two or more decimal places: 5 " 2 1 2 1 4.2360680 4.236068 4.23607 (seven places) (six places) (five places) 4.2361 4.236 4.24 (four places) (three places) (two places) Unacceptable answers would include values that are not rounded correctly, as well as values with fewer than two decimal places such as 4.2 (one decimal place) and 4 (no decimal place). 3. When the solution to a problem involving radicals has two |
or more steps, no radical should be rounded until the very last step. For example, to find the value of 3 3 first multiply the calculator approximation for product to two decimal places. 5 " 5, rounded to the nearest hundredth, " by 3 and then round the Correct Solution: Incorrect Solution(2.236067977) 6.708203931 6.71 3(2.236067977) 3(2.24) 6.72 5 The solution, 6.72, is incorrect because the rational approximation of rounded too soon. To the nearest hundredth, the correct answer is 6.71." was Radicals and the Irrational Numbers 479 EXAMPLE 1 Between which two consecutive integers is Solution How to Proceed (1) Place 42 between the squares of consecutive integers: (2) Take the square root of each number: (3) Simplify terms: (4) Multiply each term of the inequality by 1: Recall that when an inequality is multiplied by a negative number, the order of the inequality is reversed. Answer EXAMPLE 2 2 42 is between 7 and 6. " 2 42? " 36 42 49 " 49 36, 6 6 " " 2 42, " 7 7 42 42 7 2 42 6 " " Is 56 " a rational or an irrational number? Solution Since 56 is a positive integer that is not a perfect square, there is no rational number that, when squared, equals 56. Therefore, is irrational. 56 " Calculator Solution STEP 1. Evaluate 56. ENTER: Β― 56 ENTER " 2nd DISPLAY STEP 2. To show that the calculator displays a rational approximation, not an exact value, show that the square of 7.483314774 does not equal 56. ENTER: 7.483314774 x2 ENTER DISPLAY Since (7.483314774)2 56, then 7.483314774 is a rational approximation for, not an exact value. 56 " Answer 56 " is an irrational number. 480 Operations With Radicals EXAMPLE 3 Calculator Solution Is 8.0656 " rational or irrational? STEP 1. Evaluate 8.0656. " 2nd ENTER: DISPLAY: Β― 8.0656 ENTER β ( STEP 2. It appears that the rational number in the calculator display is an exact value of does equal 8.0656. " 8.0656. To verify this, show that the square of 2.84 ENTER: 2.84 x2 ENTER DISPLAY |
Since (2.84)2 8.0656, then 8.0656 2.84. " Answer 8.0656 " is a rational number. If n is a positive rational number written as a terminating decimal, then n2 has twice as many decimal places as n. For example, the square of 2.84 has four decimal places. Also, since the last digit of 2.84 is 4, note that the last digit of 2.842 must be 6 because 42 16. EXAMPLE 4 Calculator Solution Approximate the value of the expression 82 1 13 2 4 a. to the nearest thousandth b. to the nearest hundredth " ENTER: 2nd Β― 8 x2 13 ) 4 ENTER DISPLAY. To round to the nearest thousandth, look at the digit in the ten-thousandth (4th) decimal place. Since this digit (9) is greater than 5, add 1 to the digit in Radicals and the Irrational Numbers 481 the thousandth (3rd) decimal place. When rounded to the nearest thousandth, 4.774964387 is approximately equal to 4.775 b. To round to the nearest hundredth, look at the digit in the thousandth (3rd) decimal place. Since this digit (4) is not greater than or equal to 5, drop this digit and all digits to the right. When rounded to the nearest hundredth, 4.774964387 is approximately equal to 4.77 Answers a. 4.775 b. 4.77 Note: It is not correct to round 4.774964387 to the nearest hundredth by rounding 4.775, the approximation to the nearest thousandth. EXERCISES Writing About Mathematics 1. a. Use a calculator to evaluate 999,999. " b. Enter your answer to part a and square the answer. Is the result 999,999? c. Is 999,999 " rational or irrational? Explain your answer. 2. Ursuline said that is an irrational number because it is the ratio of 18 which is irra- tional and 50 which is irrational. Do you agree with Ursuline? Explain why or why not. " 18 50 $ " Developing Skills In 3β12, between which consecutive integers is each given number? 3. 8 " 4. 13 " 5. 8. 52 9. 73 40 " 2 10. 125 6. 2 2 " 143 11. 7. 2 14 " 9 1 36 12 |
. " " In 13β18, in each case, write the given numbers in order, starting with the smallest. " " " 13. 2, 16. 0 " 14. 4, 17. 5, 17, 3 " 21, " 30 " 15. 18. In 19β33, state whether each number is rational or irrational. 19. 24. 29. 25 " 400 1,156 " " 20. 25. 30. 40 " 1 2 $ 951 " 21. 26. 31. 2 2 36 " 4 9 $ 6.1504 " 22. 27. 32. 2 2 " " 15, 3, 4 11, 2 23, 2 19 " " 2 54 " 0.36 " " 2,672.89 23. 28. 33. 2 150 " 0.1 " " 5.8044 482 Operations With Radicals In 34β48, for each irrational number, write a rational approximation: a. as shown on a calculator display b. rounded to four decimal places 34. 39. 44. 2 " 90 " " 28.56 35. 40. 45. 3 " 108 " " 67.25 36. 41. 46. 21 23.5 4,389 " " " 37. 42. 47. 39 88.2 123.7 " " " 38. 43. 48. " 2 " 80 115.2 " 134.53 In 49β56, use a calculator to approximate each expression: a. to the nearest hundredth b. to the nearest thousandth. 49. 7 1 7 " 2 1 53. 3 " 57. Both 50. 54. 2 1 " 19 2 6 " 3 " 8 1 8 " 130 2 4 51. 55. " " 50 1 17 27 1 4.0038 52. 56. " " and 58.01 are irrational numbers. Find a rational number n such that 58 " n 58 " 58.01. " " Applying Skills In 58β63, in each case, find to the nearest tenth of a centimeter the length of a side of a square whose area is the given measure. 58. 18 square centimeters 59. 29 square centimeters 60. 96 square centimeters 61. 140 square centimeters 62. 202 square centimeters 63. 288 square centimeters In 64β67, find the perimeter of each figure, rounded to the nearest hundredth. 64. 3 β34 5 65. β3 β7 66. β5 β5 67. β19 10 20 β |
19 12-3 FINDING THE PRINCIPAL SQUARE ROOT OF A MONOMIAL Just as we can find the principal square root of a number that is a perfect square, we can find the principal square roots of variables and monomials that represent perfect squares. β’ Since (6)(6) 36, β’ Since (x)(x) x2, β’ Since (a2)(a2) a4, β’ Since (6a2)(6a2) 36a4, then then then then 36 x2 a4 6. x (x 0). a2. 36a4 6a2. " " " " Finding the Principal Square Root of a Monomial 483 In the last case, where the square root contains both numerical and variable factors, we can determine the square root by finding the square roots of its factors and multiplying: 36a4 5 36? " " a4 5 6? a2 5 6a2 " Procedure To find the square root of a monomial that has two or more factors, write the indicated product of the square roots of its factors. Note: In our work, we limit the domain of all variables under a radical sign to nonnegative numbers. EXAMPLE 1 Find the principal square root of each monomial. Assume that all variables represent positive numbers. a. 25y2 b. 1.44a6b2 c. 1,369m10 d. 81 4 g6 Solution a. b. c. d. 25 25y2 5 A " 1.44a6b2 5 B A " 1.44 " " y2 (5)(y) 5y Answer B a6 b2 (1.2)(a3)(b) 1.2a3b Answer A " B A " B A " m10 B (37)(m5) 37m5 Answer 1,369m10 5 " 81 4 g6 5 $ 81 4 A " 1,369 A " (g6g3) 5 9 2g3 A B Answer EXERCISES Writing About Mathematics 1. Is it true that for x 0, x2 5 2x? Explain your answer. " 2. Melanie said that when a is an even integer and x 0, Melanie? Explain why or why not. xa 5 x a 2 ". Do you agree with 484 Operations With Radicals Developing Skills In 3β14, find the principal square root of each monomial. Assume that all variables |
represent positive numbers. 3. 4a2 7. 9c2 4. 49z2 8. 36y4 5. 16 25r2 9. c2d2 6. 0.81w2 10. 4x2y2 11. 144a4b2 12. 0.36m2 13. 0.49a2b2 14. 70.56b2x10 Applying Skills In 15β18, where all variables represent positive numbers: a. Represent each side of the square whose area is given. b. Represent the perimeter of that square. 15. 49c2 16. 64x2 17. 100x2y2 18. 144a2b2 19. The length of the legs of a right triangle are represented by 9x and 40x. Represent the length of the hypotenuse of the right triangle. 12-4 SIMPLIFYING A SQUARE-ROOT RADICAL, can often be simplified. To A radical that is an irrational number, such as understand this procedure, let us first consider some radicals that are rational and that numbers. We know that 400 5 20. 36 5 6 " 12 " 36 5 6 Since and then " " ". " " " Since and then " " " 16? 25 5 16? " 16? 25 5 400 5 20. 25 5 4? 5 5 20, 16? 25. " " " These examples illustrate the following property of square-root radicals: The square root of a product of nonnegative numbers is equal to the prod- uct of the square roots of these numbers. In general, if a and b are nonnegative numbers: a? b 5 " a? b " " and a? " " b 5 a? b " Now we will apply this rule to a square-root radical with a radicand that is not a perfect square. Simplifying a Square Root Radical 485 1. Express 50 as the product of 25 and 2, where 25 is the greatest perfect-square factor of 50: 2. Write the product of the two square roots: 3. Replace 25 with 5 to obtain the simplified expression with the smallest possible radicand: " 50 5 " 25? 2 " 5 25? " 2 " 5 5 2 " The expression. The simplest form is called the simplest form of of a square-root radical is one in which the radicand is an integer that has no perfect-square factor other than 1. " " 50 2 5 If the radic |
and is a fraction, change it to an equivalent fraction that has a denominator that is a perfect square. Write the radicand as the product of a fraction that is a perfect square and an integer that has no perfect-square factor other than 1. For example: 8 3 5 $ Procedure 3 3 3 8 3 5 $ 24 " To simplify the square root of a product: 1. If the radicand is a fraction, write it as an equivalent fraction with a denomi- nator that is a perfect square. 2. Find, if possible, two factors of the radicand such that one of the factors is a perfect square and the other is an integer that has no factor that is a perfect square. 3. Express the square root of the product as the product of the square roots of the factors. 4. Find the square root of the factor with the perfect-square radicand. EXAMPLE 1 Simplify each expression. Assume that y 0. Answers a. b. c. d. 150 18 " 4 " 9 2 $ 4y3 " " 25? 6 5 4 " 25? 4y3 4y2? y 5 " $ " 18 4 5 4y2? 5 5 6 5 20 2y " y " " 486 Operations With Radicals EXERCISES Writing About Mathematics 1. Does 1 3 27 5 " 9? Explain why or why not. " 2. Abba simplified the expression 192 by writing 192 5 a. Explain why 4 12 is not the simplest form of 192. " " " " b. Show how it is possible to find the simplest form of 16? 12 5 4 12. " " 192 " by starting with 4 12. " c. What is the simplest form of 192? " Developing Skills In 3β22, write each expression in simplest form. Assume that all variables represent positive numbers. 3. 8. 13. 18. 20 12 " 2 " 1 2 $ 8 " 9x 27 " 5 " 24 36 5 $ 3x3 " 48 4. 9. 14. 19. " 23. The expression 2 (1) 3 " 24. The expression (1) 8 " 25. The expression 3 (1) 54 " 26. The expression 3 (1) 9 " 4 2 " " 3 " is equivalent to (2) 4 12 " is equivalent to (2) 42 " 18 is equivalent to (2) 3 2 " is equivalent to (2) 6 " 5. |
10. 15. 20. 63 " 1 4 " 48 32 3 $ 49x5 " 6. 11. 16. 21. 98 96 " 3 4 " 2 1 8 $ 36r2s " 7. 4 12 " 80 3 2 " 15 12. 17. 22. 2 5 $ 243xy2 " (3) 4 3 " (3) 32 " (3) 9 2 " (3) 12 " (4) 16 3 " (4) 64 " (4) 3 6 " (4) 27 " In 27β30, for each expression: a. Use a calculator to find a rational approximation of the expression. b. Write the original expression in simplest radical form. c. Use a calculator to find a rational approximation of the answer to part b. d. Are the approximations in parts a and c equal? 27. 300 " 28. 180 " 29. 2 288 " 30. 1 3 252 " Addition and Subtraction of Radicals 487? Explain your answer. 31. a. Does 9 1 16 5 9 1 16 b. Is finding a square root distributive over addition? " " 32. a. Does 169 2 25 5 169 2 25? Explain your answer. b. Is finding a square root distributive over subtraction? " " " " 12-5 ADDITION AND SUBTRACTION OF RADICALS Radicals are exact values. Computations sometimes involve many radicals, as in the following example: 12 1 75 1 3 3 " " Rather than use approximations obtained with a calculator to find this sum, it may be important to express the answer as an exact value in radical form. To learn how to perform computations with radicals, we need to recall some algebraic concepts. " Adding and Subtracting Like Radicals We have learned how to add like terms in algebra: 2x 5x 3x = 10x If we replace the variable x by an irrational number, ment must be true: 3, the following state 10 3 " " Like radicals are radicals that have the same index and the same radicand. are like are like radicals and and " " 6 5 3 2 For example, radicals. 3, " 3, and " 3 " 3 7 " 3 7 " To demonstrate that the sum of like radicals can be written as a single term, we can apply the concept used to add like terms, namely, the distributive property: 2 Similarly(2 1 5 1 3) 5 " " 3 |
(10) 5 10 (6 2 1) 5 " 3 7(5) 5 5 " 3 7 " Procedure To add or subtract like terms that contain like radicals: 1. Add or subtract the coefficients of the radicals. 2. Multiply the sum or difference obtained by the common radical. 488 Operations With Radicals Adding and Subtracting Unlike Radicals Unlike radicals are radicals that have different radicands or different indexes, or both. For example: β’ β’ β’ and and 5 3 " 3 2 " 9 10 " different. 2 2 are unlike radicals because their radicands are different. " 2 are unlike radicals because their indexes are different. " 3 4 and " are unlike radicals because their radicands and indexes are The sum or difference of unlike radicals cannot always be expressed as a single term. For instance: β’ The sum of 3 and 2 2 is 3 5 1 2 5 " β’ The difference of 5 " and " 11 is 5 7 " 2. " 7 2 11. " However, it is sometimes possible to transform unlike radicals into equivalent like radicals. These like radicals can then be combined into a single term. Let us return to the problem posed at the start of this section: " " 12 1 75 25? 10 " " 3 Procedure To combine unlike radicals: 1. Simplify each radical if possible. 2. Combine like radicals by using the distributive property. 3. Write the indicated sum or difference of the unlike radicals. EXAMPLE 1 Combine: a. 6 " " n 2 2 49n " Solution a(8 1 1 2 2) 5 5(7) 5 7 " " b. 6 n 2 2 49n 5 6 " n 2 2(7(6 2 14) 5 28 n " Answers a. 7 5 " b. 28 n " Addition and Subtraction of Radicals 489 EXAMPLE 2 Alexis drew the figure at the right. Triangles ABC and CDE are isosceles right triangles. AB 5.00 centimeters and DE 3.00 centimeters. B 5.00 cm C D 3.00 cm a. Find AC and CE. b. Find, in simplest radical form, AC CE. c. Alexis measured AC CE and found the measure to be 11.25 centimeters. Find the percent of error of the measurement that Alexis made. Express the answer to the nearest hundredth of a percent. A E Solution a. Use the Pythagorean Theorem. AC2 AB2 BC2 52 52 |
25 25 50 CE2 CD2 DE2 32 32 9 9 18 CE 18 " 2 5 5 18 5 25? " " " AC 50 " 50 1 " " b. AC CE c. Error 8 2 2 11.25 " Percent of error 8 " 2 2 11.25 2 8 " Β― 2 8 2nd 8 2nd Β― 2 ENTER: ( ( ) ) ) 11.25 ) ENTER DISPLAY Multiply the number in the display by 100 to change to percent. Percent of error 0.5631089% 0.56% Answers a. AC 50 cm, CE " cm 18 " b. 8 2 " cm c. 0.56% 490 Operations With Radicals EXERCISES Writing About Mathematics 1. Compare adding fractions with adding radicals. How are the two operations alike and how are they different? 2. Marc said that 3 5 2 5 5 3. Do you agree with Marc? Explain why or why not. " " Developing Skills In 3β23, in each case, combine the radicals. Assume that all variables represent positive numbers. 8 5. 7 7. 11. 13. 15. 17. 19. 21. 23. " 3 " 15 " " " $ " " " y 2 7 y " 27 1 " 75 " 12 2 48 1 3 " " 0.2 1 0.45 3 4 1 " 1 3 $ 100b 2 64b 1 9b 3a2 1 " 12a2 " " 4. 6. 8. 10. 12. 14. 16. 18. 20. 22. 24 " 50 2 1 " 80 2 5 " 0.98 2 4 " 72 8 9 2 $ " 7a 1 28a " 3 " 3x 2 12x " x " " " a2 1 6 a 2 3 a " 0.08 1 3 1.28 " In 25β27, in each case, select the numeral preceding the correct choice. 25. The difference 5 2 2 32 is equivalent to (1) 2 " 26. The sum 3 (1) 9 10 " " " 9 (2) 2 " 8 1 6 2 " " is equivalent to (2) 72 " 27. The sum of (1) 39 " and 12 " " (2) 5 6 " 27 is equivalent to (3) 4 30 " (3) 18 10 " (3) 13 3 " (4) 5 30 " (4) 12 2 " (4) 5 3 " Applying Skills In 28 |
and 29: a. Express the perimeter of the figure in simplest radical form. b. Using a calculator, approximate the expression obtained in part a to the nearest thousandth. Multiplication of Square-Root Radicals 491 28. 29. 5β5 4β5 3β5 β27 2β3 30. On the way to softball practice, Maggie walks diagonally through a square field and a rec- tangular field. The square field has a length of 60 yards. The rectangular field has a length of 70 yards and a width of 10 yards. What is the total distance Maggie walks through the fields? 12-6 MULTIPLICATION OF SQUARE-ROOT RADICALS To find the area of the rectangle pictured at the 3 5 2 right, we multiply. We have learned by " " ab a? when a and b are nonnegative that " 2 4 numbers. To multiply, we use the com" mutative and associative laws of multiplication as follows: 3 " b 5 by " " 4 5 5 3 5 (4)(5 In general, if a and b are nonnegative numbers: 3 5 (4? 5) 4β2 5β3 2? 3 A " 5 20 xy ab " Procedure To multiply two monomial square roots: 1. Multiply the coefficients to find the coefficient of the product. 2. Multiply the radicands to find the radicand of the product. 3. If possible, simplify the result. 492 Operations With Radicals EXAMPLE 1 a. Multiply 3 6 5 and write the product in simplest radical form. " A B A 2 " B b. Check the work performed in part a using a calculator. Solution a(5) 6(2) " 12 5 15 5 15 " 4 A " 5 15(2) 5 30 Answer 5 b. To check, evaluate ENTER: 3 2nd 2nd Β― 2 ) ENTER DISPLAY. " Β― Then evaluate 30 ENTER: 30 2nd DISPLAY: 3 0 β ( 3 3 ENTER Therefore 30 B 3 " appears to be true. EXAMPLE 2 Solution Alternative Solution A Answer 12 Find the value of 2)(2) 5 4(3) 5 12 (3) 5 12 " EXAMPLE 3 Find the indicated product: Solution 3x? " 6x 5 " " 3x? 6x 5 3x? " 18x2 5 " " 6x, |
(x 0). 9x2? " " 2 5 3x Answer 2 " Multiplication of Square-Root Radicals 493 EXERCISES Writing About Mathematics 1. When a and b are unequal prime numbers, is answer. rational or irrational? Explain your ab " 2. Is 4a2 a rational number for all values of a? Explain your answer. " Developing Skills In 3β26: in each case multiply, or raise to the power, as indicated. Then simplify the result. Assume that all variables represent positive numbers. 3. 6. 9. 12. 15. 18. 21. 24. " " " 5 3? 3 " 2x? 14? 2x " 2 " 3 " 8? 7 " 24 25x 9a 4x B B A " ab. 7. 10. 13. 16. 19. 22. 25. 7? 7 " 12? 12 60? " 5 " 24? " 21 2 B 27a " A 2 t A " A " A " 5x B A " 2 B In 27β33: a. Perform each indicated operation. ber or a rational number. B 5. 8. a? a " 18? 3 " 8 11. 14 " 15 " " " 2 B y 26 3a " 17. 20. State whether the product is an irrational num- B 15x 2 B A " 26. 23. 3x B B B 27. 5 12 31. B A B A 35. Two square-root radicals that are irrational numbers are multiplied. B A 3 B 33. 32. B B " " 18 28. 3 2 32 29 " 30. 34. A A 5 8 " 11 " 1 2 B A 38 10 " 1 11 B A B 45 " B a. Give two examples where the product of these radicals is also an irrational number. b. Give two examples where the product of these radicals is a rational number. Applying Skills In 36β39, in each case, find the area of the square in which the length of each side is given. 36. 2 " 37. 2 3 " 38. 6 2 " 39. 5 3 " In 40 and 41: a. Express the area of the figure in simplest radical form. formed in part a by using a calculator. b. Check the work per- 40. β2 2β3 41. β3 2β12 494 Operations With Radicals 12-7 DIVISION OF SQUARE- |
ROOT RADICALS Since and then " " $, Since and then. 4 9, 16 25 5 4 5 16 5 4 5 25 16 25 5 " " $ " " $ 16 25. These examples illustrate the following property of square-root radicals: The square root of a fraction that is the quotient of two positive numbers is equal to the square root of the numerator divided by the square root of the denominator. In general, if a and b are positive numbers: a b 5 " " $ a b and 5 a b a b $ " " 8. In this example, notice that the " We use this principle to divide 72 by quotient of two irrational numbers is a rational number. " 5 72 8 " " 72 8 5 " 9 5 3 $ We can also divide radical terms by using the property of fractions: bd 5 a ac b? Note, in the following example, that the quotient of two irrational numbers c d is irrational: 10 2 In general, if a and b are positive, and y 0: 10? # 10 $ Procedure To divide two monomial square roots: 1. Divide the coefficients to find the coefficient of the quotient. 2. Divide the radicands to find the radicand of the quotient. 3. If possible, simplify the result. Division of Square-Root Radicals 495 EXAMPLE 1 Divide 8 by 4 48 " " 2, and simplify the quotient. Solution 8 48 4 4 " 2 5 8 4 " 48 2 5 2 $ 24 (2) B 6 A " B 5 4 6 " Answer EXAMPLE 2 Find the indicated division:, (x 0, y 0, z 0). 2x2y3z " 6y " Solution " 2x2y3z 6y " 5 1 3? $ " x2y2z 5 3 9? $ " x2y2z 1 9x2y2? $ " 3z 5 1 3xy " 3z Answer EXERCISES Writing About Mathematics 1. Ross simplified 16 81 $ by writing 5 4 9 5 2 3 16 81 " ". Do you agree with Ross? Explain why or why not. 2. Is 1 16 rational or irrational? Explain your answer. $ Developing Skills In 3β18, divide. Write the quotient in simplest form. Assume that all variables represent positive numbers. 3. 7. 11. 15. 72 4 2 " 48 " " 12 3 " " 20. |
8. 12. 16. 75 4 " 24 4 " 7 " 20 4 50 " 2 " 5. 9. 13. 17. 70 4 " 10 " 150 " b3c4 a2 a " " 6. 10. 14. 18. 14 4 2 " 21 " 40 4 " 9y 4 " 3 x3y " z 6 " 5 " y " In 19β26, state whether each quotient is a rational number or an irrational number. 19. 5 " 7 23. 9 16 " " 20. 24. 50 2 18 25 " " " " 21. 25. 18 3 " " 25 " 5 " 24 2 22. 49 " 7 " 26. 3 " 6 " 54 3 496 Operations With Radicals In 27β34, simplify each expression. Assume that all variables represent positive numbers. 27. 31. 36 49 $ 10 8 25 $ 28. 32. 3 4 $ 9 18 xy2 y $ 29. 4 33. 5 16 $ a6b5c4 a4b3c2 $ 30. 34. 8 49 $ 15 3 a5 a2b2 $ Applying Skills In 35β38, in each case, the area A of a parallelogram and the measure of its base b are given. Find the height h of the parallelogram, expressed in simplest form. 35. A 7 37. A 8 12, b 7 45, b 2 3 15 " " " " 36. A 38. A CHAPTER SUMMARY 640, b 98, b " 2 " 32 32 " " A radical, which is the root of a quantity, is written in its general form as, with an. A radical consists of a radicand, b, placed under a radical sign, " n b " index, n. 2 49 5 " " Finding the square root of a quantity reverses the result of the operation of squaring. A square-root radical has an index of 2, which is generally not written. because 72 49. In general, for nonnegative numbers b Thus, and x, 49 5 7 if and only if x2 b. " b 5 x If k is a positive number that is a perfect square, then k is a rational num- ber. If k is positive but not a perfect square, then Every positive number has two square roots: a positive root called the principal square root, and a negative root. These roots have the same absolute value: " k " is an irrational |
number. Principal Square Root Negative Square Root Both Square Roots x2 5 ZxZ " 2 x2 5 2ZxZ " 6 x2 5 6x " Finding the cube root of a number is the inverse of the operation of cubing. if and only if because 43 = 4 4 4 64. In general, 3 b 5 x " 3 64 5 4 Thus, " x3 b. and 7 " Like radicals have the same radicand and the same index. For example, 7 3. Unlike radicals can differ in their radicands ( and " indexes ( and A square root of a positive integer is simplified by factoring out the square root of its greatest perfect square. The radicand of a simplified radical, then, has no perfect square factor other than 1. When a radical is irrational, the radical expresses its exact value. Most calculators display only rational approximations of radicals that are irrational numbers. 7 2 " 2 ), in their " ), or in both ( 3 6 ). " 3 7 " 4 7 " 7 " and Review Exercises 497 Operations with radicals include: 1. Addition and subtraction: Combine like radicals by adding or subtracting their coefficients and then multiplying this result by their common radical. The sum or difference of unlike radicals, unless transformed to equivalent like radicals, cannot be expressed as a single term. 2. Multiplication and division: For all radicals whose denominators are not equal to 0, multiply (or divide) coefficients, multiply (or divide) radicands, and simplify. The general rules for these operations are as follows xy B " ab and $ VOCABULARY 12-1 Radicand β’ Radical β’ Perfect square β’ Principal square root β’ Cube root β’ Index 12-4 Simplest form of a square-root radical 12-5 Like radicals β’ Unlike radicals REVIEW EXERCISES 1. When a b $ is an integer, what is the relationship between a and b? 2. When a is a positive perfect square and b is a positive number that is not a perfect square, is ab rational or irrational? Explain your answer. " 3. Write the principal square root of 1,225. 4. Write the following numbers in order starting with the smallest: 18, " 2 2, 3. " In 5β14, write each number in simplest form. 9 25 $ 400y4 5. 9. 13. " " 6. 10. 2 49 " 180 " 0.01m16 " |
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