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| > 7. Write the solution set in interval notation and graph the solution set. Solution Solution Solution Solution Solution 2x – 3 < –7 or 2x < –4 x < –2 2x – 3 > 7 WWWWWrite the inequality as a disjunction rite the inequality as a disjunction rite the inequality as a disjunction rite the inequality as a disjunction ri...
solution set in interval notation. 26. |a – 8| > 1 27. |t + 2| > 8 28. |7a| β‰₯ 14 29. |4j| β‰₯ 16 30. c 3 > 6 31. c 12 β‰₯ 1 3 32. |x + 9| > 7 33. |x – 11| > 12 34. |3x – 7| > 13 35. |5y + 11| > 21 36. |4m + 9| β‰₯ 11 37. |7b – 8| β‰₯ 13 38. 3 4 x βˆ’ 3 β‰₯ 5 39. 2 1 x βˆ’ 7 β‰₯ 3 40. |3(x – 2) + 7| β‰₯ 8 41. |4(3 + x)| β‰₯ 13 42. |2(3x –...
.1 if you’ve forgotten what the β€œΒ»β€œ sign means. Section 3.4 Section 3.4 Section 3.4 β€” Absolute Value Inequalities Section 3.4 Section 3.4 161161161161161 Chapter 3 Investigation Mailing Pacacacacackakakakakagggggeseseseses Mailing P Mailing P Mailing P Mailing P Mailing Pacacacacackakakakakagggggeseseseses Mailing P Ma...
you think will be most efficient. (Assume the box does not need any tabs to stick it together.) ound Up ound Up RRRRRound Up ound Up ound Up ound Up ound Up RRRRRound Up ound Up ound Up In general, problems that use the words β€œmaximum,” β€œminimum,” β€œlimits,” or β€œtolerance” often mean that you need to use inequalities t...
it out: The coordinates of the origin are (0, 0). Plotting coordinates isn’t anything new to you β€” you’ve had lots of practice in earlier grades. This Topic starts right at the beginning though, to remind you of the earlier work. e Used to Locate Pte Pte Pte Pte Points on a Plane oints on a Plane oints on a Plane e Us...
Mor y Need to Plot Mor ou Ma YYYYYou Ma ou Ma ou May Need to Plot Mor han One P y Need to Plot Mor ou Ma You can often join up several plotted points to create the outline of a shape. Example Example Example Example Example 22222 Draw a coordinate plane and plot and label the points M (3, 4), A (–3, 4), C (–3, –4), an...
ises 3–7. 3. What are the coordinates of point A? 4. What are the coordinates of point B? 5. What are the coordinates of point C? 6. What are the coordinates of point D? 7. What are the coordinates of point E? D –4 –2 4 2 0 –2 –4 C A 2 B 4 E In Exercises 8–11, plot the coordinates on a coordinate plane. 8. The origin 9...
r Quadr the Plane the Plane ants of ants of Quadr Quadr ants of the Plane Quadrants of the Plane ants of Quadr the Plane Quadr There are four main regions of the coordinate plane β€” they’re divided up by the x- and y-axes. This Topic is about spotting where on the coordinate plane points lie. our Quadrantsantsantsantsan...
> 0). Guided Practice In Exercises 1–6, name the quadrant or axis where each point is located. Justify your answers. 1. (1, 0) 4. (0, –5) In Exercises 7–12, a > 0, k > 0, m < 0, v < 0, and p Ε’ R. Name the quadrant or axis where each point is located. Justify your answers. 7. (a, k) 10. (p, 0) 3. (–4, –1) 6. (–p, p) 8....
points lie on. Justify your answer. a) (0, 3) b) (12, 2) c) (45, 0) Solution Solution Solution Solution Solution a) (0, 3) is on the y-axis, since x = 0. b) (12, 2) isn’t on an axis, since neither x nor y is 0. c) (45, 0) is on the x-axis, since y = 0. Guided Practice In Exercises 19–24, let t > 0. State which axis, i...
g Students g ph a linear ph a linear Students g tion tion equa equa tion and compute the xequation equa tion equa and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4). What it means for you: You’ll learn a...
1 y an Equationtiontiontiontion y an Equa y an Equa A Line Can Be Described b A Line Can Be Described b A Line Can Be Described by an Equa y an Equa A Line Can Be Described b A Line Can Be Described b Check it out: At the point (3, 5), x = 3. You can check that (3, 5) is of the form (x, 2x – 1) by substituting 3 for x....
= 0 14. x = 0 Draw the graphs for Exercises 15–16 on coordinate grids spanning –6 to 6 on the x-axis and –6 to 6 on the y-axis. 15. Draw the graph of the set of all points (x, y) such that x = y. 16. Draw the graph of the set of points (x, y) such that x = 5 and y Ε’ R. Describe the line you have drawn. 17. If x Ε’ {–1,...
the two lines defined in Exercise 6 and the relationship between them. 8. Plot the two lines whose points are defined by (x, –2x + 1) and (x, –2x – 2). 9. Describe the two lines defined in Exercise 8 and the relationship between them. 10. Draw two lines whose points are defined by (x, 2x + 1) and (x, –0.5x + 1). 11. G...
equa equa tion and compute the xequation equa tion equa and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4). What it means for you: You’ll check whether lines are horizontal or vertical. Key words: horizo...
of x are the same (–3 for the line x = –3, 2 for the line x = 2). PPPPPoints on a Horiz oints on a Horiz oints on a Horiz ontal Line ha ontal Line ha ontal Line havvvvve the Same e the Same e the Same -Coor -Coor -Coordinadinadinadinadinatetetetete oints on a Horizontal Line ha e the Same y-Coor oints on a Horiz ontal...
coordinate plane. 1. x = 8 and x = 0 3. x = 4 and x = –6 5. y = –4 and y = –6 2. x = –3 and x = 1 4. y = 0 and y = 3 6. y = 2 and y = –3 In Exercises 7–12, write the equation for each line on the graph. y 4 3 2 1 –6 –5 –4 –3 –2 –1 0 –1 –2 –3 –4 –5 –6 –7 –8 7. 654321 x 8. 9. 10. y 20 18 16 14 12 10 8 6 4 2 –12–10–8–6 –...
verify You already dealt with lines in Topics 4.1.3 and 4.1.4. In this Topic you’ll see a formal definition relating ordered pairs to a line β€” and you’ll also learn how to show that points lie on a particular line. Graphs of Linear Equations are Straight Lines An equation is linear if the variables have an exponent of...
4. Solution a) 2 – 3(–3) = 11 2 + 9 = 11 11 = 11 Substitute 2 for x and –3 for y A true statement So the point (2, –3) lies on the graph of x – 3y = 11, since (2, –3) satisfies the equation x – 3y = 11. b) If (–1, 1) lies on the line, 2(–1) + 3(1) = 4. But 2(–1) + 3(1) = –2 + 3 = 1 Since 1 Ο€ 4, (–1, 1) does not lie on...
point lies on the graph of 6x – 6y = 24. 9. (4, 0) 11. (100, 96) 10. (1, –3) 12. (–400, –404) 13. Explain in words why (2, 31) is a point on the line x = 2 but not a point on the line y = 2. 14. Determine whether the point (3, 4) lies on the lines 4x + 6y = 36 and 8x – 7y = 30. Round Up Round Up You can always substit...
y = x + 3. Choose two values of x, then draw a table to help you find the y-values. x 2– 4 y (x, y) y = x 3+ 3+2–= 1= )1,2–( y = x 3+ 3+4= 7= )7,4( When you plot the graph, the line should be straight. y-axis –3 (4, 7) y = x – (1, 4) 1 2 3 4 5 6 x-axis 1 –1 –2 –3 (–2, 1) –6 –5 –4 –3 –2 Checkx, y) = (1, 4) (1, 4) lies ...
) and (2, 4) 3. (0, 0) and (2, 6) 2. (–1, –1) and (1, 3) 4. (0, –2) and (1, 1) Graph and label the lines of the equations in Exercises 5–16. 5. x + y = 8 7. 2x + y = –3 9. –3x + y = –6 11. 2x – y = –14 13. 8x + 4y = 24 15. 3x – 9y = –27 6. y – x = 10 8. 5x + y = –12 10. –10x + y = 21 12. 6x + 2y = 18 14. 12x – 4y = 8 1...
to calculate unknown values: Example 1 (2k, 3) is a point on the line x – 3y = 7. Find k. Justify your answer. Solution x – 3y = 7 2k – 3(3) = 7 2k – 9 = 7 2k – 9 + 9 = 7 + 9 2k = 16 k = 2 2 k = 8 16 2 Substituting 2k for x and 3 for y Addition Property of Equality Division Property of Equality So the point on the lin...
οΏ½ k 6. Find the coordinates of the point in Exercise 7. 180 Section 4.2 β€” Lines Example 3 The point (1, 3) lies on the line bx + y = 6. Find the value of b. Solution Here you have to use the coordinates to identify the equation. The question is different, but the method is the same. bx + y = 6 b(1 Guided Practice 7. Th...
(1, 1) lies on the line bx – 2by = 4. Find b. 12. The point (–3, 6) lies on the line 4x + 6ky = 24. Find k. 13. The point (4k, 2k) lies on the line 2x – 6y = 12. Find k and the coordinates of the point. Round Up Round Up This Topic’s really just an application of the method you learned in Topic 4.2.1. Once you’ve subs...
x + 6y < 4). What it means for you: You’ll learn about x- and y-intercepts and how to compute them from the equation of a line. Key words: intercept linear equation 182 Section 4.2 β€” Lines Check it out: Always write the x-intercept as a point, not just as the value of x where the graph crosses the x-axis. For example, ...
) is the y-intercept of –2x – 3y = –9. Example 4 Find the y-intercept of the line 3x + 4y = 24. Solution Let x = 0, then solve for y: 3x + 4y = 24 3(0) + 4y = 24 0 + 4y = 24 4y = 24 y = 6 So (0, 6) is the y-intercept of 3x + 4y = 24. Guided Practice In Exercises 9–16, find the y-intercept. 9. 4x – 6y = 24 10. 5x + 8y =...
, use the graph below to help you reach your answer6 –5 –4 –3 –2 –1 0 –1 1 2 3 4 5 x 6 –2 –3 –4 –5 –6 18. Find the x- and y-intercepts of line n. 19. Find the x-intercept of line p. 20. Find the y-intercept of line r. 21. Explain why line p does not have a y-intercept. 22. Explain why line r does not have an x-intercep...
y. Draw a set of axes and plot the two intercepts. Draw a straight line through the points. Check your line by plotting a third point. Example 1 Draw the graph of 5x + 3y = 15 by computing the intercepts. Solution x-intercept: 5x + 3(0) = 15 5x + 0 = 15 5x = 15 x = 3 y-intercept: 5(0) + 3y = 15 0 + 3y = 15 3y = 15 y =...
y = – 2 5 x – 2 16. y = 18 Section 4.2 β€” Lines 187 Independent Practice Draw graphs of the lines using the x- and y-intercepts in Exercises 1–6. 1. x-intercept: (–3, 0) 2. x-intercept: (1, 0) 3. x-intercept: (4, 0) 4. x-intercept: (–6, 0) 5. x-intercept: (–1, 0) 6. x-intercept: (2, 0) y-intercept: (0, 2) y-intercept: ...
.0:7.0:7.0:7.0: Students verify that a point lies on a line, given an equation of the line. le to derivvvvveeeee le to deri Students are ae ae ae ae abbbbble to deri le to deri Students ar Students ar le to deri Students ar Students ar tions tions linear equa linear equa tions by using the linear equations tions linear...
189189189 a Line a Line ula to Find the Slope of Use the Fororororormmmmmula to Find the Slope of ula to Find the Slope of Use the F Use the F a Line ula to Find the Slope of a Line a Line ula to Find the Slope of Use the F Use the F Example Example Example Example Example 11111 Find the slope of the line that passes t...
. Check it out: It doesn’t matter which point you call (x1, y1) and which you call (x2, y2) β€” choose whichever makes the math easier. This time the line has a negative slope, meaning it goes β€œdownhill” from left to right. Here the slope is –2, which means that the line goes 2 units down for every 1 unit across. w a Gra...
Solution Even though one pair of coordinates contains a variable, k, you still use the slope formula in exactly the same way as before. m = y x 2 2 βˆ’ βˆ’, which means that But the slope is 3, so 1 2 k + 2 = 3 fi 2k + 1 = 6 fi 2k = 5 fi k = 5 2 Guided Practice Find the slope m of the lines through the points below. 17. (7, ...
and (0, 2) 4. (–5, 2) and (–1, 3) 3. (4, 4) and (1, 0) 5. (3, –3) and (7, 3) In Exercises 6–10, find the slope of the line through each of the points. 6. (–3, 5) and (2, 1) 8. (2, 3) and (4, 3) 10. (2s, 2t) and (s, 3t) 7. (0, 4) and (–4, 0) 9. (6d, 2) and (4d, –1) In Exercises 11–15, you’re given two points on a line ...
F The point-slope formula is a really useful way of calculating the equation of a straight line. aight Line aight Line a Str a Str tion of tion of ula to Find the Equa Use the Fororororormmmmmula to Find the Equa ula to Find the Equa Use the F Use the F aight Line a Straight Line tion of a Str ula to Find the Equation...
2 5 TTTTTopicopicopicopicopic 4.3.24.3.2 4.3.24.3.2 4.3.2 California Standards: 7.0:7.0:7.0:7.0:7.0: Students verify that a point lies on a line, given an equation of the line. Students are ae ae ae ae abbbbble to deri le to derivvvvveeeee le to deri le to deri Students ar Students ar Students ar le to deri Students a...
Slope Step 2: Write the equation y – y1 = m(x – x1) (x – 3) y – (–2 4y + 8 = –7(x – 3) fi 4y + 8 = –7x + 21 fi 4y + 7x = 13 (x – 3) Guided Practice Write the equation of the line that passes through the given pair of points. 9. (–1, 0) and (3, –4) 11. (–5, 7) and (3, 9) 13. (8, 7) and (–7, –5) 15. (3, 1) and (5, 4) 17. ...
3) and (4, –1) 9. (3, 8) and (4, 4) 11. (–6, 9) and (–4, –6) 13. (–4, –8) and (–5, 4) 15. (10, 5) and (4, 6) 8. (–1, 6) and (7, 5) 10. (4, –7) and (–3, 5) 12. (4, –9) and (–3, –9) 14. (–8, 3) and (8, 4) 16. (0, 0) and (–4, –6) 17. The points (5, 6) and (8, 7) lie on a line. Find the equation of this line. 18. The line...
4 allel Lines allel Lines PPPPParararararallel Lines allel Lines allel Lines PPPPParararararallel Lines allel Lines allel Lines allel Lines allel Lines California Standards: 8.0:8.0:8.0:8.0:8.0: Students under stand stand Students under Students under stand Students understand stand Students under allel allelallel par ...
the same plane that never intersect are called _________________ lines. 2. To determine if two lines are parallel you can look at their _________________. 3. Prove that the line f defined by y – 3 = 2 3 (x – 4) is parallel to line g defined by y – 6 = 2 3 (x + 1). t Havvvvve Defined Slopes e Defined Slopes e Defined S...
absolutely accurate, you should check that this isn’t the case before you say the lines are parallel. You can do this by finding the equations of the lines and comparing them. 198198198198198 Section 4.4 β€” More Lines Section 4.4 Section 4.4 Section 4.4 Section 4.4 Guided Practice 4. Show that line a, which goes throug...
it out: This is the equation of the line through (–1, 4) that is parallel to the straight line joining (5, 7) and (–6, –8). [x – (–1)] fi y – 4 = 15 11 fi 11y – 44 = 15(x + 1) fi 11y – 44 = 15x + 15 Equation: 11y – 15x = 59 Section 4.4 β€” More Lines Section 4.4 Section 4.4 Section 4.4 Section 4.4 199199199199199 Guided Pr...
the line through (–5, 1) and (–5, 5). 8. Determine if the line through the points (–2, 3) and (–2, –2) is parallel to the line through (1, 7) and (–6, 7). 9. Find the equation of the line through (1, –2) that is parallel to the line joining the points (–3, –1) and (8, 7). 10. Find the equation of the line through (–5,...
thr line tha line tha ough a ough a t passes thr line that passes thr ough a t passes through a line tha t passes thr line tha ough a gigigigigivvvvven point. en point. en point. en point. en point. What it means for you: You’ll work out the slopes of perpendicular lines and you’ll test if two lines are perpendicular....
run formula: Don’t forget: See Topic 4.3.1 for the β€œrise over run” formula. Slope of A = m1 = Slope of B = m2 = 1 2 = 2 4 βˆ’4 2 = –2 4 2 2 4 B A 1 2 is the negative reciprocal of –2, so A and B must be perpendicular. Section 4.4 β€” More Lines Section 4.4 Section 4.4 Section 4.4 Section 4.4 201201201201201 Guided Practic...
A is – 5 8, find the slope of B. 8. Lines R and T are perpendicular. If R has slope – 7 11, what is the slope of T? 9. The slope of l1 is –0.8. The slope of l2 is 1.25. Determine whether l1 and l2 are perpendicular. inding SSSSSlopes lopes lopes inding pendicular by y y y y FFFFFinding inding pendicular b ines are e e...
(3, –4) that is perpendicular to the line through the points (–7, –3) and (–3, 8). 14. Determine the equation of the line through (6, –7) that is perpendicular to the line through the points (8, 2) and (–1, 8). 15. Find the equation of the line through (4, 5) that is perpendicular to the line –3y + 4x = 6. Independent...
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cept form or not. 1. y = 3x + 7 3. y – 3 = 2(x – 4) 2. 3x + 4y = 7 4. y – 8 = 3(x – 4) 5. y = 3 2 x + 18 6. y = –4x – 1 t Easy to Plot Graaaaaphsphsphsphsphs t Easy to Plot Gr m Makes es es es es IIIIIt Easy to Plot Gr t Easy to Plot Gr m Mak he Slope-Intercececececept Fpt Fpt Fpt Fpt Forororororm Mak m Mak he Slope-In...
, find the slope. 8. In the equation y = –x + 10, find the slope. 9. In the equation y = 2x + 5, find the y-intercept. 10. In the equation y = 7b – 3, find the y-intercept. In Exercises 11–18, plot each equation on a graph. 11. y = 2x + 3 12. y = x – 6 13. y = –7x – 8 15. y = – 1 2 x + 6 17. y = 3 4 x 14. y = – 1 3 x –...
in slope-intercept form. The slope, m = – the y-coordinate of the y-intercept, b = – C B. A B and Example Example Example Example Example 44444 Determine the slope and y-intercept of the line 2x – 3y = 9. Solution Solution Solution Solution Solution Step 1: Solve the given equation for y. 2x – 3y = 9 –3y = –2x + Now y...
1 3 x + 5 13. y = x + 2 15. y = 2x 12. y = – 1 3 x – 6 14. y = –x + 2 In Exercises 16–20, write the equations of the lines in slope-intercept form. 4 3 1 2 16. A line with slope 17. A line with slope 18. 4x + 2y = 8 20. 3x – 4y = –16 that passes through the point (0, 4) that passes through the point (0, –2) 19. 6x – 3...
ugh a gi ough a gi thrthrough a gi thrthr en point. en point. ough a gi thr What it means for you: You’ll learn how to tell whether lines are parallel or perpendicular by looking at the slope-intercept form. Key words: parallel perpendicular reciprocal Don’t forget: See Topic 4.3.1 for more on the slope of a line. Don’...
οΏ½ 3(y + 4) = 2(x – 4) fi 3y + 12 = 2x – 8 fi 33333y – 2x = –20 Guided Practice 1. Give an example of a line that is parallel to y = 1 2 x + 1. 2. Is the line y = 4 5 x – 2 parallel to the line y = 4 5 x + 6? Explain. 3. Find the equation of the line through (–4, 3) that is parallel to the line y = 3x + 9. 4. Find the equ...
x – x1) fi y – (–4) = 3(x – 2) fi y + 4 = 3x – 6 fi y – 3x = –10 Guided Practice 5. Give an example of a line that’s perpendicular to the line y = 6x. 6. Is the line y = 4x + 2 perpendicular to the line y = – 1 4 x + 4? Explain your answer. 7. Find the equation of the line through (–2, 0) that is perpendicular to the line...
line through (3, 5) that’s parallel to 3x – 7y = –21. 15. The line through (4, –3) that’s parallel to 3x – 4y = 16. 16. The line through (–2, 6) that’s parallel to 6x – 10y = –20. 17. The line through (0, 6) that’s perpendicular to 2x + y = 18. 18. The line through (–3, –5) that’s perpendicular to 3x – 6y = –24. 19. T...
by Inequalities by Inequalities by Inequalities by Inequalities by Inequalities by Inequalities Just like with equations, you can graph inequalities on the coordinate plane. The only tricky bit is showing whether the solution set is above or below the line. This Topic will show you how. vides the Plane into TTTTThrhrh...
also e also e also h the reeeeegiongiongiongiongion h the r le to sketcetcetcetcetch the r h the r le to sk aaaaabbbbble to sk le to sk h the r le to sk defined by linear inequality defined by linear inequality defined by linear inequality defined by linear inequality defined by linear inequality (e(e(e(e(e.g.g.g.g.g....
1, a β€œ>” sign is needed to make it a true statement. So (0, 2) satisfies the inequality y > 2x + 1. Therefore the inequality that defines the shaded region is y > 2x + 1. Guided Practice In Exercises 1–2, state the inequality that defines the shaded region on each of the graphs. 1. –6 6 5 4 3 2 1 0 –1 –1 –2 –3 –4 –5 –...
graph below. Now test whether the point (0, 0) satisfies the inequality. Substitute x = 0 and y = 0 into the inequality. 6x – 3y < 9 0 – 0 < 9 0 < 9 β€” This is a true statement. Therefore (0, 0) lies in the region 6x – 3y < 9 β€” so shade the region containing (0, 0). y 6x – 6 –5 –4 –3 –2 0 –1 –1 –2 –3 –4 –5 –6 y-axis 1 ...
. 3. y > 0.5x + 2 5. y + x > –2 7. –2y + 3x > 6 4. y + 2x < 0 6. 4x + 3y < 12 8. y < –x + 3 In Exercises 9–14, show whether the given point is a solution of –5x + 2y > –8. 9. (0, 0) 12. (2, 1) 11. (–3, 9) 14. (–15, 13) 10. (6, –3) 13. (39, –36) Section 4.5 Section 4.5 Section 4.5 β€” Inequalities Section 4.5 Section 4.5 ...
y < 3 4 x + 6 15. y < – 2 5 x – 2 17. x > 0 19. x + 2y > 8 21. 4x – 6y > 24 14. y < 4 5 x + 4 16. y < 1 18. x – 4y > 8 20. 4x + 3y < –12 22. 5x + 8y < 24 ound Up ound Up RRRRRound Up ound Up ound Up RRRRRound Up ound Up ound Up ound Up ound Up Well, that was quite a long Topic, with lots of graphs. Inequality graphs a...
strict inequality border region Don’t forget: Remember, Β£ means β€œless than or equal to,” β‰₯ means β€œgreater than or equal to.” gions gions s of R R R R Reeeeegions s of s of BorBorBorBorBorderderderderders of gions gions s of s of R R R R Reeeeegions BorBorBorBorBorderderderderders of gions gions s of s of gions s of gi...
x – – – – – y £££££ –2 –2 –2 –2 –2 –6 –5 –4 –3 –2 y-axis 22222x – – – – – y β‰₯β‰₯β‰₯β‰₯β‰₯ –2 –2 –2 –2 –2 0 –1 –1 –2 –3 –4 –5 –6 Set of points bbbbbeloeloeloeloelowwwww and on and on and on the line β€” all and on and on these points satisfy the inequality 22222x – – – – – y β‰₯β‰₯β‰₯β‰₯β‰₯ –2 –2 –2 –2 –2. x-axis Section 4.5 Section 4.5 Se...
statement. y-axis Therefore (0, 0) doesn’t lie in the region y Β£ –x – 5 β€” so shade the region that doesn’t contain (0, 0). 1 2 3 4 5 6 x-axis –6 –5 –4 –3 –2 y Β£ x– – 5 4 3 2 1 0 –1 –1 –2 –3 –4 –5 –6 –7 –8 Guided Practice In Exercises 1–4, show whether the given point is in the solution set of –2x + 3y Β£ –15. 1. (0, 0)...
false statement. y-axis Therefore (0, 0) doesn’t lie in the region 4y – 3x β‰₯ 12 β€” so shade the region that doesn’t contain (0, 0). 4 – 3 y x β‰₯ 12 8 7 6 5 4 3 2 1 –6 –5 –4 –3 –2 0 –1 –1 –2 –3 1 2 3 4 5 6 x-axis Section 4.5 Section 4.5 Section 4.5 β€” Inequalities Section 4.5 Section 4.5 219219219219219 Check it out: Afte...
Β£ 0 14. y β‰₯ –3x 16. x + 4y < 4 18. 4x – 3y β‰₯ 8 19. Show whether (5, 4) is in the solution set of 4x – 3y Β£ 8. 20. Show whether (–4, 2) is in the solution set of 2x + y > –6. 220220220220220 Section 4.5 Section 4.5 Section 4.5 β€” Inequalities Section 4.5 Section 4.5 Independent Practice In Exercises 1–4, show whether th...
including Β£ and β‰₯ signs will always have a solid line. Section 4.5 Section 4.5 Section 4.5 β€” Inequalities Section 4.5 Section 4.5 221221221221221 TTTTTopicopicopicopicopic 4.5.34.5.3 4.5.34.5.3 4.5.3 California Standards: 9.0:9.0:9.0:9.0:9.0: Students solve a system of two linear equations in two variables algebraical...
ions Defined by Mor han One Linear Inequality y Mor gions Defined b A system of linear inequalities is made up of two or more linear inequalities that contain the same variables. For example, 3x + 2y > 6 and 4x – y < 5 are linear inequalities both containing the variables x and y. An ordered pair (x, y) is a solution o...
β‰₯ –25 and y – x Β£ –5. Solution Solution Solution Solution Solution First line: 5y + 3x = –25 fi 5y = –3x – 25,x ) ( y x 0 5 y –= y –= 3 5 3 5 x 5– x 5– 5–)0(–= 3 5 5–)5(–= 3 5 5–= )5–,0( 8–= )8–,5( The border line y = – 3 5 and is a solid line. x – 5 goes through the points (0, –5) and (5, –8), Second line: y – x = –,x...
y < 6 and y – 2x < 2 2. y – x β‰₯ 4 and 2x + y Β£ 5 In each of Exercises 3–6, use a set of axes spanning from –6 to 6 on the x- and y-axes. For each exercise, shade the region containing all solution points for both inequalities. 3. y < –x + 4 and y < x 5. y < x and y < –x – 2 4. y < x + 2 and y > –2x + 5 6. y Β£ 0.5x + 3 ...
Example 3 contin Example 3 contin Example 3 contin Example 3 contin Second line: Two points on this line are (3, 2) and (1, –2). m1 = βˆ’ βˆ’ – y1 = m(x – x1) fi y – 2 = 2(x – 3) fi y – 2 = 2x – 6 fi y = 2x – 4 So the equations of the two border lines are y = 1 3 x + 1 and y = 2x – 4. Choose a point in the shaded region, for...
4 –5 –6 Independent Practice In Exercises 1–4, use the graph opposite to determine if the given point is included in the solution set. 1. (0, 0) 2. (2, 6) 3. (6, 2) 4. (4, –2) –6 –5 –4 –3 –2 5. Determine whether the point (–4, –3) lies in the solution region of both 3x – 4y Β£ 2 and x – 2y β‰₯ 1. y-axis 6 5 4 3 2 1 0 –1 –...
ound Up ound Up ound Up ound Up RRRRRound Up ound Up ound Up When you’re graphing a system of linear inequalities, don’t forget that you still have to pay attention to whether the lines should be solid or dashed. 226226226226226 Section 4.5 Section 4.5 Section 4.5 β€” Inequalities Section 4.5 Section 4.5 Chapter 4 Inves...
to pass through the origin? Why? What is the slope of the best-fit line? What does the slope of your graph represent? Find the equation of the best-fit line. a) How old would you expect a tree to be if its diameter is 23 cm? b) What would you expect the diameter of a 6-year-old tree to be? Extension It’s difficult to ...
phing Method phing Method he Gr he Graaaaaphing Method TTTTThe Gr phing Method phing Method he Gr he Gr phing Method he Gr phing Method California Standards: Students solve ae ae ae ae a 9.0:9.0:9.0:9.0:9.0: Students solv Students solv Students solv Students solv system of tw tw tw tw two linear o linear o linear syst...
system are often called simultaneous equations because any solution has to satisfy the equations simultaneously (at the same time). The equations can’t be solved independently of one another. y Graaaaaphing phing phing y Gr y Gr tions b tions b Equa Equa Solving Systems of Solving Systems of phing tions by Gr Equation...
fi 7 = 7 β€” True statement –2x + y = –1 fi –2(–1) + (–3) = –1 fi –1 = –1 β€” True statement Therefore x = –1, y = –3 is the solution of the system of equations. 230230230230230 Section 5.1 Section 5.1 Section 5.1 β€” Systems of Equations Section 5.1 Section 5.1 Guided Practice Solve each system of equations in Exercises 1–6 b...
8 13. y = –x + 6 and x – y = –4 14. x – y = 1 and x + y = –3 15. x + y = 1 and x – 2y = 1 16. 2x + y = –8 and 3x + y = –13 ound Up ound Up RRRRRound Up ound Up ound Up RRRRRound Up ound Up ound Up ound Up ound Up There’s something very satisfying about taking two long linear equations and coming up with just a one-coo...
solutions, and even integer solutions if the scale of your graph is small. e Graaaaaphsphsphsphsphs e Gr he Substitution Method Doesn’’’’’t Int Int Int Int Invvvvvolvolvolvolvolve Gr e Gr he Substitution Method Doesn TTTTThe Substitution Method Doesn he Substitution Method Doesn e Gr he Substitution Method Doesn The s...
system of equations. It’s a good idea to check that the solution is correct by substituting it into the original equations. 2x – 3y = 7 fi 2(–1) – 3(–3) = 7 fi –2 + 9 = 7 fi 7 = 7 β€” True statement –2x + y = –1 fi –2(–1) + (–3) = –1 fi 2 – 3 = –1 fi –1 = –1 β€” True statement The solution makes both of the original equations t...
233233 TTTTTopicopicopicopicopic 5.1.35.1.3 5.1.35.1.3 5.1.3 California Standards: Students solve ae ae ae ae a 9.0:9.0:9.0:9.0:9.0: Students solv Students solv Students solv Students solv system of tw tw tw tw two linear o linear o linear system of system of o linear system of o linear system of tions in two vo vo vo ...
2y + x = 8 2y + 2x = 10 Solution Solution Solution Solution Solution Step 1: Graph both equations in the same coordinate plane. Line of first equation: 2y + x = 8 fi 2y = – The line goes through the points (0, 4) and (2, 3). Line of second equation: 2y + 2x = 10 fi 2y = –2x + 10 fi y = –x + 5 x 0 3 y 5 2 The line goes th...
y = –6 y = 3 Step 3: Substitute 3 for y in an equation to find x. Equation 3 is the best one to use here because x is already isolated β€” so you don’t have to do any rearranging. x = –2y + 8 x = –2(3) + 8 x = 2 So x = 2, y = 3, or (2, 3), is the solution of the system of equations. Check by substituting the solution in ...
by substitution. 8. 2x – y = 8 and y = 4 10. y = 2x – 1 and x + y = 5 12. 6x + y = – 2 and 4x – 3y = 17 13. 4x – 5y = 0 and 4x – 3y = 8 9. x + y = 0 and y = –3x 11. y = 3x and 2x + 3y = 44 ound Up ound Up RRRRRound Up ound Up ound Up RRRRRound Up ound Up ound Up ound Up ound Up The graphing method works well with simp...
have no solutions. Key words: inconsistent point of intersection parallel substitution system of linear equations Check it out: Equations with identical slopes, but different intercepts, result in parallel lines. So you can tell if the system is inconsistent without even drawing the graphs, by rearranging the equation...
ivvvvves a F es a F he Substitution Method Gi TTTTThe Substitution Method Gi he Substitution Method Gi tement alse Statement es a False Sta tement alse Sta es a F he Substitution Method Gi Here’s the same problem you saw in Example 1, but this time using the substitution method. Example Example Example Example Example ...
2x = 10 – 4y ound Up ound Up RRRRRound Up ound Up ound Up ound Up ound Up RRRRRound Up ound Up ound Up From this Topic you’ve seen that it doesn’t matter whether you use the graphing or substitution method β€” if the system is inconsistent, you won’t get any solutions. 238238238238238 Section 5.1 Section 5.1 Section 5.1...
Many Solutions A Dependent System Has Infinitely Many Solutions A Dependent System Has Infinitely Many Solutions Some systems of two linear equations have an infinite number of solutions β€” in other words, there are an infinite number of points (x, y) that satisfy both of the equations in the system. Every solution of ...
but this time using the substitution method. Example Example Example Example Example 22222 Solve this system of equations by substitution: y + x = 4 (Equation 1) 2y + 2x = 8 (Equation 2) Solution Solution Solution Solution Solution Step 1: Rearrange one equation so that one of the variables is expressed in terms of th...
ound Up ound Up ound Up It would be crazy to try to list the solutions to dependent systems, because there are infinitely many. You can actually rearrange the equations and show that they’re really saying the same thing. 240240240240240 Section 5.1 Section 5.1 Section 5.1 β€” Systems of Equations Section 5.1 Section 5.1 ...
β€” Furβ€” Furβ€” Furβ€” Furβ€” Further Examples ther Examples ther Examples β€” Furβ€” Furβ€” Furβ€” Furβ€” Further Examples ther Examples ther Examples ther Examples ther Examples There’s nothing new to learn in this Topic β€” just practice at using the methods you’ve learned to solve systems of equations. tions tions Equa Equa t Solving...
x = –27 x = 3 Step 3: Substitute 3 for x in Equation 3 to find y. y = 3x – 5 y = 3(3 So x = 3, y = 4 is the solution of the system of equations. Check by substituting the solution in the original equations. y – 3x = –5 4 – 3(3) = –5 4 – 9 = –5 –5 = – 5 β€” True statement 3x – 4y = –7 3(3) – 4(4) = –7 9 – 16 = –7 –7 = –7 ...
+ 18y = –1 11. 6x + 6y = –6 and 11y – x = 25 12. –4x + 7y = 15 and 8x – 14y = 37 13. –7x + 5y = 20 and 14x – 10y = –40 14. 5x + 6y = 5 and –5x + 3y = 1 15. –4x + 7y = 9 and 5x – 2y = –18 16. 5x – y = –15 and –3x + 9y = 93 17. –4x + 9y = 57 and 5x + 5y = 75 18. –2x + 5y = 33 and 10x – 25y = 69 19. –3x + 4y = 41 and –12...
sketch the solution sets. What it means for you: You’ll learn about the elimination method and then use it to solve systems of linear equations in two variables. Key words: system of linear equations elimination method Check it out: The size (absolute value) is the important thing here. It doesn’t matter if the coeffi...
a variable. In this case, the coefficients of x are opposites of each other, so adding the equations will eliminate x. –3y + 4x = 11 + 10y – 4x = 10 7y = 21 fi y = 3 244244244244244 Section 5.2 Section 5.2 Section 5.2 β€” The Elimination Method Section 5.2 Section 5.2 Example 1 continueduedueduedued Example 1 contin Exam...
of one variable the same in both equations. To make the coefficients of variable b the same, multiply the first equation by 2 and the second equation by 3. 2(5a + 3b = 19) fi 10a + 6b = 38 3(3a + 2b = 12) fi 9a + 6b = 36 Now you have two equations that have the same coefficient of b. Section 5.2 Section 5.2 Section 5.2 ...
Section 5.2 Example 3 continueduedueduedued Example 3 contin Example 3 contin Example 3 contin Example 3 contin Step 2: Add these equations together to eliminate x. 3y + 6x = –6 + 12y – 6x = 21 15y = 15 fi y = 1 Step 2: Substitute 1 for y in one of the original equations and solve for x. 4y – 2x = 7 4(1) – 2x = 7 4 – 2...
out: Subtracting the second equation from the first means that there are no negative terms in the resulting equation. Check it out: You could also substitute into one of the equations in which the fractional coefficients have been converted to integers. Example 4 continueduedueduedued Example 4 contin Example 4 contin...