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three variables. An infinite number of solutions can result from several situations. The three planes could be the same, so that a solution to one equation will be the solution to the other two equations. All three equations could be different but they intersect on a line, which has infinite solutions. Or two of the e...
equations. Figure 11.18 Does the generic solution to a dependent system always have to be written in terms of x? No, you can write the generic solution in terms of any of the variables, but it is common to write it in terms of x and if needed x and y. 11.11 Solve the following system. x + y + z = 7 3x βˆ’ 2y βˆ’ z = 4 x +...
4, 2, βˆ’61 and (4, 4, βˆ’1) 87. βˆ’x βˆ’ y + 2z = 3 5x + 8yβˆ’3z = 4 βˆ’x + 3yβˆ’5z = βˆ’5 and (4, 1, βˆ’7) the following exercises, For substitution. solve each system by 3xβˆ’4y + 2z = βˆ’15 2x + 4y + z = 16 2x + 3y + 5z = 20 88. 89. This content is available for free at https://cnx.org/content/col11758/1.5 5xβˆ’2y + 3z = 20 2xβˆ’4yβˆ’3z = βˆ’9 ...
+ 3z = βˆ’14 βˆ’16yβˆ’24z = βˆ’112 101. 102. 103. 104. 5xβˆ’3y + 4z = βˆ’1 βˆ’4x + 2yβˆ’3z = 0 βˆ’x + 5y + 7z = βˆ’11 x + y + z = 0 2x βˆ’ y + 3z = 0 x βˆ’ z = 0 3x + 2yβˆ’5z = 6 5xβˆ’4y + 3z = βˆ’12 4x + 5yβˆ’2z = 15 x + y + z = 0 2x βˆ’ y + 3z = 0 x βˆ’ z = 1 105. 3x βˆ’ 1 2 y βˆ’ z = βˆ’ 1 2 4x + 106. 107. 6xβˆ’5y + 6z = 38 4x βˆ’ 74 = βˆ’ 13 10 z = βˆ’ 7 20 z = βˆ’...
βˆ’0.3z = βˆ’0.03 0.5xβˆ’0.5yβˆ’0.3z = 0.13 0.4xβˆ’0.1yβˆ’0.3z = 0.11 0.2xβˆ’0.8yβˆ’0.9z = βˆ’0.32 120. 0.5x + 0.2yβˆ’0.3z = 1 0.4xβˆ’0.6y + 0.7z = 0.8 0.3xβˆ’0.1yβˆ’0.9z = 0.6 1244 121. 122. 0.3x + 0.3y + 0.5z = 0.6 0.4x + 0.4y + 0.4z = 1.8 0.4x + 0.2y + 0.1z = 1.6 0.8x + 0.8y + 0.8z = 2.4 0.3xβˆ’0.5y + 0.2z = 0 0.1x + 0.2y + 0.3z = 0.6 Extensio...
-half the number of cats, and there are 20 more cats than dogs, how many of each animal are at the shelter? Your roommate, Sarah, offered to buy groceries for 132. you and your other roommate. The total bill was $82. She forgot to save the individual receipts but remembered that your groceries were $0.05 cheaper than h...
137. were three types of coins: pennies, nickels, and dimes. If the bag contained the same number of nickels as dimes, how many of each type of coin was in the bag? 130. consisting of children, parents, and grandparents, At a family reunion, there were only blood relatives, in 138. This content is available for free a...
BMW increased by 8%, the Jeep by 5%, and the Toyota by 12%. If the price of last year’s Jeep was $7,000 less than the price of last year’s BMW, what was the price of each of the three cars last year? A recent college graduate took advantage of his 139. business education and invested in three investments immediately a...
, production and consumption in 2001,” accessed April 6, 2014, http://scaruffi.com/politics/oil.html. 2. β€œOil reserves, production and consumption in 2001,” accessed April 6, 2014, http://scaruffi.com/politics/oil.html. 3. β€œOil reserves, production and consumption in 2001,” accessed April 6, 2014, http://scaruffi.com/p...
use for nonlinear systems. We solve one equation for one variable and then substitute the result into the second equation to solve for another variable, and so on. There is, however, a variation in the possible outcomes. Intersection of a Parabola and a Line There are three possible types of solutions for a system of ...
1 x = 1 x βˆ’ (1) = βˆ’1 x = 0 The solutions are (1, 2) and (0, 1), which can be verified by substituting these (x, y) values into both of the original equations. See Figure 11.21. Figure 11.21 Could we have substituted values for y into the second equation to solve for x in Example 11.17? Yes, but because x is squared in ...
2 = 5 x2 + 9x2 βˆ’30x + 25 = 5 10x2 βˆ’30x + 20 = 0 1250 Chapter 11 Systems of Equations and Inequalities Now, we factor and solve for x. 10(x2 βˆ’ 3x + 2) = 0 10(x βˆ’ 2)(x βˆ’ 1) = 0 x = 2 x = 1 Substitute the two x-values into the original linear equation to solve for y. y = 3(2)βˆ’5 = 1 y = 3(1)βˆ’5 = βˆ’2 The line intersects the ...
.24 Example 11.19 Solving a System of Nonlinear Equations Representing a Circle and an Ellipse Solve the system of nonlinear equations. x2 + y2 = 26 (1) 3x2 + 25y2 = 100 (2) Solution Let’s begin by multiplying equation (1) by βˆ’3, and adding it to equation (2). ( βˆ’ 3)(x2 + y2) = ( βˆ’ 3)(26) βˆ’ 3x2 βˆ’ 3y2 = βˆ’ 78 3x2 + 25y2 ...
a solid line. The This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 11 Systems of Equations and Inequalities 1253 Figure 11.26 (a) an example of y > a; (b) an example of y β‰₯ a; (c) an example of y < a; (d) an example of y ≀ a Given an inequality bounded by a parabola, sketch a graph. 1...
of nonlinear inequalities is similar to graphing a system of linear inequalities. The difference is that our graph may result in more shaded regions that represent a solution than we find in a system of linear inequalities. The solution to a nonlinear system of inequalities is the region of the graph where the shaded ...
Elimination (http://openstaxcollege.org/l/ nonlinelim) This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 11 Systems of Equations and Inequalities 1257 11.3 EXERCISES Verbal 148. Explain whether a system of two nonlinear equations can have exactly two solutions. What about exactly three...
= 1 y = 20x2 βˆ’1 x2 + y2 = 1 y = βˆ’ x2 2x3 βˆ’ x2 = y y = 1 2 βˆ’ x 9x2 + 25y2 = 225 (xβˆ’6)2 + y2 = 1 x4 βˆ’ x2 = y x2 + y = 0 2x3 βˆ’ x2 = y x2 + y = 0 For the following exercises, use any method to solve the nonlinear system. 171. x2 + y2 = 9 y = 3 βˆ’ x2 172. x2 βˆ’ y2 = 9 x = 3 1258 173. 174. 175. 176. 177. 178. 179. 180. 181. 1...
graph the inequality. 193. y β‰₯ e x y ≀ ln(x) + 5 194. y ≀ βˆ’ log(x) y ≀ e x For the following exercises, find the solutions to the nonlinear equations with two variables. 195. 196. 197. 198. 4 x2 + 1 x2 βˆ’ 2 5 y2 = 24 y2 + 4 = 0 6 x2 βˆ’ 1 x2 βˆ’ 6 y2 = 8 y2 = 1 8 1 x2 βˆ’ xy + y2 βˆ’2 = 0 x + 3y = 4 x2 βˆ’ xyβˆ’2y2 βˆ’6 = 0 x2 + y2 ...
( x ), where Q( x ) has repeated linear factors. 11.4.3 Decompose P( x ) Q( x ), where Q( x ) has a nonrepeated irreducible quadratic factor. 11.4.4 Decompose P( x ) Q( x ), where Q( x ) has a repeated irreducible quadratic factor. Earlier in this chapter, we studied systems of two equations in two variables, systems o...
original rational expressions, which were added or subtracted, had one of the linear factors as the denominator. In other words, using the example above, the factors of x2 βˆ’ xβˆ’6 are (xβˆ’3)(x + 2), the denominators of the decomposed rational expression. So we will rewrite the simplified form as the sum of individual fra...
2)(xβˆ’1) Solution We will separate the denominator factors and give each numerator a symbolic label, like A, B, or C. 3x (x + 2)(xβˆ’1) = A (x + 2) + B (xβˆ’1) Multiply both sides of the equation by the common denominator to eliminate the fractions: ⎑ (x + 2)(xβˆ’1) ⎣ 3x (x + 2)(xβˆ’1) ⎑ ⎀ ⎦ = (x + 2)(xβˆ’1) ⎣ A (x + 2) ⎀ ⎦ + (x...
+ 0 = A βˆ’6 βˆ’3 2 = A We obtain the same values for A and B using either method, so the decompositions are the same using either method. 3x (x + 2)(xβˆ’1) = 2 (x + 2) + 1 (xβˆ’1) Although this method is not seen very often in textbooks, we present it here as an alternative that may make some partial fraction decompositions ...
11.23 Decomposing with Repeated Linear Factors Decompose the given rational expression with repeated linear factors. βˆ’x2 + 2x + 4 x3 βˆ’4x2 + 4x Solution The denominator factors are x(xβˆ’2)2. To allow for the repeated factor of (xβˆ’2), the decomposition will include three denominators: x, (xβˆ’2), and (xβˆ’2)2. Thus, βˆ’x2 + 2x...
) + C = 2 βˆ’ Thus, βˆ’x2 + 2x + 4 x3 βˆ’4x2 + 4x = 1 x βˆ’ 2 (xβˆ’2) + 2 (xβˆ’2)2 11.17 Find the partial fraction decomposition of the expression with repeated linear factors. 6xβˆ’11 (xβˆ’1)2, Where Q(x) Has a Nonrepeated Irreducible Decomposing P(x) Q(x) Quadratic Factor So far, we have performed partial fraction decomposition with...
linear expressions such as A1 x + B1, A2 x + B2, etc., for the numerators of each quadratic factor in the denominator. P(x) Q(x) = A ax + b + A1 x + B1 ⎝a1 x2 + b1 x + c1 βŽ› ⎞ ⎠ + A2 x + B2 ⎝a2 x2 + b2 x + c2 βŽ› ⎞ ⎠ + β‹… β‹… β‹… + An x + Bn βŽ› ⎝an x2 + bn x + cn ⎞ ⎠ 2. Multiply both sides of the equation by the common denomin...
x + 2) + (Bx + C)(x + 3) Notice we could easily solve for A by choosing a value for x that will make the Bx + C term equal 0. Let x = βˆ’3 and substitute it into the equation. 8x2 + 12x βˆ’ 20 = A(x2 + x + 2) + (Bx + C)(x + 3) 8( βˆ’ 3)2 + 12( βˆ’ 3) βˆ’ 20 = A(( βˆ’ 3)2 + ( βˆ’ 3) + 2) + (B( βˆ’ 3) + C)(( βˆ’ 3) + 3) 16 = 8A A = 2 126...
(A + B)x2 + (A + 3B + C)x + (2A + 3C) So the system of equations would be: A + B = 8 A + 3B + C = 12 2A + 3C = βˆ’20 Find the partial fraction decomposition of the expression with a nonrepeating irreducible quadratic 11.18 factor. 5x2 βˆ’6x + 7 ⎞ βŽ› ⎝x2 + 1 (xβˆ’1) ⎠ Decomposing P(x) Q(x) When Q(x) Has a Repeated Irreducible...
factors, and linear expressions such as A1 x + B1, A2 x + B2, etc., for the numerators of each quadratic factor in the denominator written in increasing powers, such as P(x) Q(x) = A ax + b + A1 x + B1 (ax2 + bx + c) + A2 x + B2 (ax2 + bx + c)2 + β‹― + An + Bn (ax2 + bx + c) n 2. Multiply both sides of the equation by t...
side. x4 + x3 + x2 βˆ’ x + 1 = A(x4 + 2x2 + 1) + Bx4 + Bx2 + Cx3 + Cx + Dx2 + Ex = Ax4 + 2Ax2 + A + Bx4 + Bx2 + Cx3 + Cx + Dx2 + Ex Now we will collect like terms. x4 + x3 + x2 βˆ’ x + 1 = (A + B)x4 + (C)x3 + (2A + B + D)x2 + (C + E)x + A Set up the system of equations matching corresponding coefficients on each side of t...
Equations and Inequalities 1269 11.4 EXERCISES Verbal 206. Can any quotient of polynomials be decomposed into at least two partial fractions? If so, explain why, and if not, give an example of such a fraction Can you explain why a partial fraction decomposition 207. is unique? (Hint: Think about it as a system of equa...
234. 235. 5x + 14 2x2 + 12x + 18 5x2 + 20x + 8 2x(x + 1)2 4x2 + 55x + 25 5x(3x + 5)2 54x3 + 127x2 + 80x + 16 2x2 (3x + 2)2 x3 βˆ’5x2 + 12x + 144 ⎞ x2 βŽ› ⎝x2 + 12x + 36 ⎠ 1270 Chapter 11 Systems of Equations and Inequalities For the following exercises, find the decomposition of the partial fraction for the irreducible no...
15 ⎞ βŽ› ⎝x2 + 4 ⎠ 2 249. 250. This content is available for free at https://cnx.org/content/col11758/1.5 251. 252. 253. 254. 255. x3 βˆ’ x2 + xβˆ’1 ⎞ βŽ› ⎝x2 βˆ’3 ⎠ 2 x2 + 5x + 5 (x + 2)2 x3 + 2x2 + 4x 2 ⎞ βŽ› ⎝x2 + 2x + 9 ⎠ x2 + 25 ⎞ βŽ› ⎝x2 + 3x + 25 ⎠ 2 2x3 + 11x + 7x + 70 βŽ› ⎞ ⎝2x2 + x + 14 ⎠ 2 256. 5x + 2 xβŽ› ⎞ ⎝x2 + 4 ⎠ 2 257....
credit: β€œSD Dirk,” Flickr) Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season. Table 11.1 shows the needs of both teams. Wildcats Mud Cats Goals Balls Jerseys Table 11.1 6 30 14 10 24 20 A goal costs $300; a ball costs $10; and a jersey costs $30. How can we ...
matrix is a matrix consisting of one row with dimensions 1 Γ— n. [a11 a12 a13] A column matrix is a matrix consisting of one column with dimensions m Γ— 1. a11 ⎑ a21 ⎒ ⎣ a31 ⎀ βŽ₯ ⎦ A matrix may be used to represent a system of equations. In these cases, the numbers represent the coefficients of the variables in the syste...
It is also associative. A + B = C such that ai j + bi j = ci j A βˆ’ B = D such that ai j βˆ’ bi j = di j A + B = B + A (A + B) + C = A + (B + C) Example 11.27 Finding the Sum of Matrices Find the sum of A and B, given Solution Add corresponding entries. A = a b ⎑ ⎣ c d ⎀ ⎦ and ⎀ ⎦ ⎀ ⎦ Example 11.28 This content is availa...
⎣ ⎦ 0 βˆ’1 0 b. Subtract the corresponding entries5 ⎀ ⎀ ⎑ 10 βˆ’2 6 2 βˆ’10 βˆ’2 βŽ₯ βŽ₯ βˆ’ ⎒ 0 βˆ’12 βˆ’4 12 10 14 ⎦ ⎣ ⎦ 2 4 βˆ’2 2 βˆ’2 2 βˆ’ 6 βˆ’10 βˆ’ 10 βˆ’2 + 2 ⎀ ⎑ βŽ₯ ⎒ 12 + 12 10 + 4 14 βˆ’ 0 ⎣ ⎦ βˆ’4 βˆ’20 0 ⎀ ⎑ βŽ₯ ⎒ 24 14 14 ⎦ ⎣ βˆ’4 4 9 11.20 Add matrix A and matrix B. A = ⎑ ⎀ 6 2 ⎒ βŽ₯ and B = 0 1 ⎣ ⎦ 1 βˆ’3 ⎀ ⎑ 3 βˆ’2 βŽ₯ ⎒ 5 1 ⎦ ⎣ 3 βˆ’4 Finding Scala...
16 ⎒ ⎣ (0.15)16 (0.15)27 (0.15)34 (0.15)34 ⎀ βŽ₯ βŽ₯ ⎦ = ⎑ 2.25 ⎒ 2.4 ⎣ 2.4 4.05 5.1 5.1 ⎀ βŽ₯ ⎦ We must round up to the next integer, so the amount of new equipment needed is Adding the two matrices as shown below, we see the new inventory amounts This means ⎑ 15 ⎒ 16 ⎣ 16 ⎀ βŽ₯ + ⎦ 27 34 34 ⎑ ⎑ 18 ⎒ 19 ⎣ 19 ⎀ βŽ₯ ⎦ 32 40 40 C2...
⎒ 0 βˆ’1 ⎣ 4 ⎀ 0 βŽ₯ and B = 2 ⎦ 3 βˆ’6 Solution First, find 3A, then 2B. ⎑ βˆ’1 ⎒ ⎣ 2 0 βˆ’3 0 ⎀ 1 βŽ₯ 2 ⎦ 1 βˆ’4 3A = = 3 β‹… 1 3(βˆ’2(βˆ’1) 3 β‹… 2 βŽ₯ ⎒ 3(βˆ’66 ⎑ ⎀ ⎒ βŽ₯ 0 βˆ’3 6 ⎣ ⎦ 9 βˆ’18 12 2B = = 2(βˆ’12 ⎒ ⎣ 2 β‹… 1 ⎀ βŽ₯ 2(βˆ’3) 2 β‹… 2 βŽ₯ 2 β‹… 1 2(βˆ’4) ⎦ ⎀ 2 4 βŽ₯ 0 βˆ’6 4 ⎦ 2 βˆ’8 0 Now, add 3A + 2B. This content is available for free at https://cnx.org/c...
to a specific pattern as outlined below. The process of matrix multiplication becomes clearer when working a problem with real numbers. To obtain the entries in row i of AB, we multiply the entries in row i of A by column j in B and add. For example, the product of AB given matrices A and B, where the dimensions of A ...
column 2 of B; row 2 of A times column 3 of B. When complete, the product matrix will be AB = a11 β‹… b11 + a12 β‹… b21 + a13 β‹… b31 ⎑ ⎒ a21 β‹… b11 + a22 β‹… b21 + a23 β‹… b31 ⎣ a11 β‹… b12 + a12 β‹… b22 + a13 β‹… b32 a21 β‹… b12 + a22 β‹… b22 + a23 β‹… b32 a11 β‹… b13 + a12 β‹… b23 + a13 β‹… b33 a21 β‹… b13 + a22 β‹… b23 + a23 β‹… b33 ⎀ βŽ₯ ⎦ Propertie...
the number of columns in A matches the number of rows in B. The resulting product will be a 2Γ—2 matrix, the number of rows in A by the number of columns in B. AB = ⎑ βˆ’4 ⎣ 2 ⎀ 5 βˆ’1 βŽ₯ 0 ⎦ 3 = = 4(5) + 0(βˆ’4) + 5(2) βˆ’1(5) + 2(βˆ’4) + 3(2) βˆ’1(βˆ’1) + 2(0) + 3(3) ⎑ ⎀ ⎣ ⎦ 4(βˆ’1) + 0(0) + 5(3) ⎑ βˆ’7 10 ⎣ 30 11 ⎀ ⎦ b. The dimensions...
3 so the product is undefined. Example 11.35 Using Matrices in Real-World Problems Let’s return to the problem presented at the opening of this section. We have Table 11.3, representing the equipment needs of two soccer teams. 1282 Chapter 11 Systems of Equations and Inequalities Wildcats Mud Cats 6 30 14 Goals Balls ...
98 βŽ₯. ⎒ 74 ⎦ ⎣ 42 βˆ’75 25 βˆ’56 βˆ’67 Solution On the matrix page of the calculator, we enter matrix A above as the matrix variable [A], matrix B above as the matrix variable [B], and matrix C above as the matrix variable [C]. On the home screen of the calculator, we type in the problem and call up each matrix variable as n...
D = ⎒ 8 92 ⎦ ⎣ 12 6 ⎀ ⎑ 10 14 βŽ₯, E = ⎒ 7 2 ⎦ ⎣ 5 61 ⎀ ⎑ 6 12 ⎦, F = ⎣ 14 5 ⎑ ⎀ 0 9 ⎒ βŽ₯ 78 17 ⎣ ⎦ 15 4 270. A + B 271. C + D 272. A + C 273. B βˆ’ E 274. C + F 275. D βˆ’ B For the following exercises, use the matrices below to perform scalar multiplication. A = ⎀ ⎑ 4 6 ⎦, B = ⎣ 13 12 ⎑ 3 9 ⎒ 21 12 ⎣ 0 64 ⎀ βŽ₯, C = ⎦ ⎀ ⎑ 16...
290. 2C + B 291. 3D + 4E 292. Cβˆ’0.5D 293. 100Dβˆ’10E For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: A2 = A β‹… A ) A = ⎑ βˆ’10 20 ⎣ 5 25 ⎀ ⎦, B = ⎀ ⎑ 40 10 ⎦, C = ⎣ βˆ’20 30 ⎑ βˆ’1 ⎒ ⎣ ⎀ 0 βŽ₯ 0 βˆ’1 ⎦ 0 1 294...
0 ⎑ ⎒ ⎣ 0.5 4 ⎀ 9 βŽ₯, B = 1 8 βˆ’3 ⎦ 5 ⎑ 0.5 3 0 ⎒ βˆ’4 1 6 ⎣ 8 7 2 ⎀ βŽ₯, ⎀ βŽ₯ ⎦ 314. AB 315. BA 316. CA 1286 Chapter 11 Systems of Equations and Inequalities 11.6 | Solving Systems with Gaussian Elimination Learning Objectives In this section, you will: 11.6.1 Write the augmented matrix of a system of equations. 11.6.2 Writ...
system of equations such as has a coefficient matrix and is represented by the augmented matrix ⎀ ⎑ 3 4 ⎦ ⎣ 4 βˆ’2 3x βˆ’ 2xβˆ’3z = 2 ⎑ ⎀ 3 βˆ’1 βˆ’3 2 Notice that the matrix is written so that the variables line up in their own columns: x-terms go in the first column, y-terms in the second column, and z-terms in the third colu...
1 βˆ’3 βˆ’5 2 βˆ’5 βˆ’4 4 5 ⎀ βŽ₯ ⎦ | βˆ’2 5 6 Solution When the columns represent the variables x, y, and z, ⎑ ⎒ ⎣ βˆ’3 1 βˆ’3 βˆ’5 2 βˆ’5 βˆ’4 4 5 | βˆ’2 5 6 ⎀ βŽ₯ β†’ ⎦ x βˆ’ 3y βˆ’ 5z = βˆ’ 2 2x βˆ’ 5y βˆ’ 4z = 5 βˆ’3x + 5y + 4z = 6 11.23 Write the system of equations from the augmented matrix. 1 βˆ’1 1 ⎑ ⎒ 2 βˆ’9 | Performing Row Operations on a Matrix Now...
(Notation: Ri + cR j) Each of the row operations corresponds to the operations we have already learned to solve systems of equations in three variables. With these operations, there are some key moves that will quickly achieve the goal of writing a matrix in rowechelon form. To obtain a matrix in row-echelon form for ...
Solving a 2Γ—2 System by Gaussian Elimination Solve the given system by Gaussian elimination. 2x + 3y = 6 x βˆ’ y = 1 2 Solution First, we write this as an augmented matrix. 1290 Chapter 11 Systems of Equations and Inequalities We want a 1 in row 1, column 1. This can be accomplished by interchanging row 1 and row 2. ⎑ 3...
1 + R2 = R2 β†’ ⎑ ⎒ | Example 11.41 Solving a Dependent System Solve the system of equations. 3x + 4y = 12 6x + 8y = 24 Solution Perform row operations on the augmented matrix to try and achieve row-echelon form | 12 24 βˆ’ 1 2 R2 + R1 = R1 β†’ ⎑ 6 ⎣ 0 R1 ↔ R2 β†’ 8 ⎀ ⎦ 0 | 24 0 ⎑ 0 ⎣ 6 0 24 ⎀ ⎦ 0 8 | The matrix ends up with a...
οΏ½οΏ½ ⎒1 βˆ’3 0 ⎒ 0 ⎣ 4 1 βˆ’2 1 0 11.25 Write the system of equations in row-echelon form. x βˆ’ 2y + 3z = 9 βˆ’ x + 3y = βˆ’ 4 2x βˆ’ 5y + 5z = 17 Solving a System of Linear Equations Using Matrices We have seen how to write a system of equations with an augmented matrix, and then how to use row operations and backsubstitution to o...
57 | ⎀ 8 βŽ₯ βˆ’15 ⎦ 57 βˆ’ 1 57 R3 = R3 β†’ ⎑ ⎒1 βˆ’12 The last matrix represents the equivalent system. x βˆ’ y + z = 8 y βˆ’ 12z = βˆ’15 z = 1 Using back-substitution, we obtain the solution as (4, βˆ’3, 1). Example 11.44 Solving a Dependent System of Linear Equations Using Matrices Solve the following system of linear equations usi...
available for free at https://cnx.org/content/col11758/1.5 Chapter 11 Systems of Equations and Inequalities 1295 11.26 Solve the system using matrices. x + 4y βˆ’ z = 4 2x + 5y + 8z = 15 x + 3yβˆ’3z = 1 Can any system of linear equations be solved by Gaussian elimination? Yes, a system of linear equations of any size can ...
variables. Let x = the amount invested at 10.5% interest, and y = the amount invested at 12% interest. x + y = 12,000 0.105x + 0.12y = 1,335 As a matrix, we have Multiply row 1 by βˆ’0.105 and add the result to row 2. ⎑ 1 1 ⎣ 0.105 0.12 12,000 1,335 ⎀ ⎦ | Then, ⎑ 1 1 ⎣ 0 0.015 12,000 75 ⎀ ⎦ | 0.015y = 75 y = 5,000 So 12...
βˆ’20,000 1 0.03 0 1 0.03 βˆ’2 1 βˆ’1 | 10,000 βˆ’3 | 0 βˆ’2 βˆ’3| 3| 10,000 ⎀ βŽ₯ βŽ₯ 9,000 βŽ₯ βˆ’2,000 ⎦ ⎀ 10,000 βŽ₯ 9,000 βŽ₯ βˆ’20,000 ⎦ The third row tells us βˆ’ 1 3 z = βˆ’2,000; thus z = 6,000. The second row tells us y + 4 3 z = 9,000. Substituting z = 6,000, we get (6,000) = 9,000 y + 4 3 y + 8,000 = 9,000 y = 1,000 The first row tells...
not. Explain how to write that system of equations. 336. 337. 338. 326. Is there only one correct method of using row operations on a matrix? Try to explain two different row operations possible to solve the augmented matrix ⎑ 3 9 ⎣ 1 βˆ’2 ⎀ ⎦. 6 | 0 Can a matrix whose entry is 0 on the diagonal be 327. solved? Explain ...
x + y = 14 347. βˆ’4xβˆ’3y = βˆ’2 3xβˆ’5y = βˆ’13 348. βˆ’5x + 8y = 3 10x + 6y = 5 349. 3x + 4y = 12 βˆ’6xβˆ’8y = βˆ’24 350. βˆ’60x + 45y = 12 20xβˆ’15y = βˆ’4 351. 11x + 10y = 43 15x + 20y = 65 Chapter 11 Systems of Equations and Inequalities 1299 352. 2x βˆ’ y = 2 3x + 2y = 17 y βˆ’ z = βˆ’3 ⎑ βˆ’0.1 0.3 βˆ’0.1 ⎒ 0.1 βˆ’0.4 0.2 ⎣ 0.7 0.6 0.1 ⎀ 0.2 βŽ₯ 0....
+ 2y + zβˆ’3 3 = 5 371. xβˆ’1 4 βˆ’ y + 1 4 + 3z = βˆ’βˆ’2 2 βˆ’ z = 4 = 1 372. 373. xβˆ’3 4 x + 5 2 + βˆ’ yβˆ’1 3 y + 5 2 + + 2z = βˆ’βˆ’3 10 x + 5 4 xβˆ’1 4 + y + 3 2 yβˆ’1 8 y + 4 2 βˆ’ βˆ’2z = 3 + z = 3 2 + 3z = 3 2 Chapter 11 Systems of Equations and Inequalities ice cream, find out the percentage of ice cream sales each individual ice cream ...
50, how much was in each account after the year passed? You invested $2,300 into account 1, and $2,700 into 378. account 2. If the total amount of interest after one year is $254, and account 2 has 1.5 times the interest rate of account 1, what are the interest rates? Assume simple interest rates. Bikes’R’Us manufactur...
? What is the best method to solve this problem? There are several ways we can solve this problem. As we have seen in previous sections, systems of equations and matrices are useful in solving real-world problems involving finance. After studying this section, we will have the tools to solve the bond problem using the ...
βˆ’1, the multiplicative inverse of a matrix A. 1302 Chapter 11 Systems of Equations and Inequalities Example 11.48 Showing That the Identity Matrix Acts as a 1 Given matrix A, show that AI = IA = A2 5 Solution Use matrix multiplication to show that the product of A and the identity is equal to the product of the identit...
free at https://cnx.org/content/col11758/1.5 Chapter 11 Systems of Equations and Inequalities 1303 11.28 Show that the following two matrices are inverses of each other. A = 1 ⎀ ⎑ 4 ⎦, B = ⎣ βˆ’1 βˆ’3 ⎀ ⎑ βˆ’3 βˆ’4 ⎦ ⎣ 1 1 Finding the Multiplicative Inverse Using Matrix Multiplication We can now determine whether two matrices...
row 2, column 2 equal to the corresponding entry of the identity. Using row operations, multiply and add as follows: (βˆ’2)R1 + R2 = R2. Add the two equations and solve for d. 1bβˆ’2d = 0 R1 2bβˆ’3d = 1 R2 1304 Chapter 11 Systems of Equations and Inequalities Once more, back-substitute and solve for b. 1bβˆ’2d = 0 0 + 1d = 1 ...
Find the Multiplicative Inverse of Matrix A Use the formula to find the multiplicative inverse of A = ⎀ ⎑ 1 βˆ’2 ⎦ ⎣ 2 βˆ’3 Solution Using the formula, we have ⎑ βˆ’3 2 ⎣ βˆ’2 1 ⎀ ⎦ Aβˆ’1 = = = 1 (1)(βˆ’3) βˆ’ (βˆ’2)(2) ⎀ ⎑ βˆ’3 2 ⎦ ⎣ βˆ’2 1 1 βˆ’3 + 4 ⎀ ⎑ βˆ’3 2 ⎦ ⎣ βˆ’2 1 Analysis We can check that our formula works by using one of the other...
to obtain the inverse. Given a 3Γ—3 matrix augment A with the identity matrix | ⎀ βŽ₯ ⎦ | To begin, we write the augmented matrix with the identity on the right and A on the left. Performing elementary row operations so that the identity matrix appears on the left, we will obtain the inverse matrix on the right. We will ...
= I and Aβˆ’1 A = I. AAβˆ’1 = = = ⎀ βŽ₯ ⎦ ⎑ βˆ’1 ⎒ βˆ’2 βˆ’(βˆ’1) + 3(βˆ’1) + 1(6) 2(1) + 3(0) + 1(βˆ’2) 2(0) + 3(1) + 1(βˆ’3) ⎀ ⎑ βŽ₯ ⎒ 3(βˆ’1) + 3(βˆ’1) + 1(6) 3(1) + 3(0) + 1(βˆ’2) 3(0) + 3(1) + 1(βˆ’3) βŽ₯ ⎒ 2(βˆ’1) + 4(βˆ’1) + 1(6) 2(1) + 4(0) + 1(βˆ’2) 2(0) + 4(1) + 1(βˆ’3 ⎀ βŽ₯ ⎦ 1308 Chapter 11 Systems of Equations and Inequalities Aβˆ’2 βˆ’3 2 4 1 βˆ’1(2) ...
equations with the same number of equations as variables as To solve a system of linear equations using an inverse matrix, let A be the coefficient matrix, let X be the variable matrix, and let B be the constant matrix. Thus, we want to solve a system AX = B. For example, look at the following system of equations. AX ...
equations, write the coefficient matrix A, the variable matrix X, and the constant matrix B. Then Multiply both sides by the inverse of A to obtain the solution. AX = B βŽ› ⎝Aβˆ’1⎞ ⎠AX = ⎑ ⎠A⎀ βŽ› ⎝Aβˆ’1⎞ ⎣ ⎦X = IX = X = βŽ› ⎝Aβˆ’1⎞ ⎠B βŽ› ⎝Aβˆ’1⎞ ⎠B βŽ› ⎝Aβˆ’1⎞ ⎠B βŽ› ⎝Aβˆ’1⎞ ⎠B If the coefficient matrix does not have an inverse, does that ...
⎑ y ⎣ ⎑ 5 ⎣ 7 (Aβˆ’1)AX = (Aβˆ’1)B ⎀ ⎑ ⎑ 11 βˆ’4 3 4 11 ⎑ ⎀ 11(5) + (βˆ’8)4(5) + 3(7) ⎑ ⎀ βˆ’ ⎣ Can we solve for X by finding the product BAβˆ’1? No, recall that matrix multiplication is not commutative, so Aβˆ’1 B β‰  BAβˆ’1. Consider our steps for solving the matrix equation. βŽ› ⎝Aβˆ’1⎞ ⎠AX = ⎑ ⎠A⎀ βŽ› ⎝Aβˆ’1⎞ ⎣ ⎦X = IX = βŽ› ⎝Aβˆ’1⎞ ⎠B βŽ› ⎝Aβˆ’1⎞...
2. Add row 1 to row 3. Multiply row 2 by βˆ’3 and add to row 1. Multiply row 3 by 5. Multiply row 3 by 1 5 and add to row 1 19 ⎒ 19 ⎒ 5 ⎣ 0 0 1 5 ⎑ 56 15 ⎒ βˆ’4 βˆ’11 βˆ’41 ⎣ βˆ’1 βˆ’3 βˆ’11 56 5 ⎑ ⎒ ⎒ βˆ’4 βˆ’11 βˆ’41 ⎣ βˆ’1 βˆ’3 βˆ’11 1 βˆ’3 βˆ’11 56 5 19 56 ⎒ 5 ⎒ 0 1 19 ⎒ βˆ’ 11 5 4 5 1 5 βˆ’ 11 19 ⎒ 5 ⎣ 0 0 1 βˆ’2 βˆ’3 1 4 5 1 1 0 0 5 ⎀ βŽ₯ βŽ₯ ⎦ 1312 Cha...
2z = βˆ’2 Given a system of equations, solve with matrix inverses using a calculator. 1. Save the coefficient matrix and the constant matrix as matrix variables [A] and [B]. 2. Enter the multiplication into the calculator, calling up each matrix variable as needed. 3. If the coefficient matrix is invertible, the calcula...
not. Explain what condition is necessary for an inverse to exist. Can you explain whether a 2Γ—2 matrix with an entire 387. row of zeros can have an inverse? Can a matrix with an entire column of zeros have an 388. inverse? Explain why or why not. Can a matrix with zeros on the diagonal have an 389. inverse? If so, fin...
οΏ½οΏ½ ⎒ 1 βˆ’3 4 ⎦ ⎣ βˆ’2 βˆ’4 βˆ’5 ⎀ ⎑ 1 9 βˆ’2 7 3 ⎀ ⎑ 1 βˆ’2 βŽ₯ ⎒ βˆ’4 8 βˆ’12 ⎦ ⎣ ⎀ βŽ₯ ⎦ For the following exercises, solve the system using the inverse of a 2 Γ— 2 matrix. 411. Chapter 11 Systems of Equations and Inequalities 1315 5x βˆ’ 6y = βˆ’ 61 4x + 3y = βˆ’ 2 412. 8x + 4y = βˆ’100 3xβˆ’4y = 1 413. 3xβˆ’2y = 6 βˆ’x + 5y = βˆ’2 414. 5xβˆ’4y = βˆ’5 4x ...
2y = 2.3 428. 429. 430 + 11 2 5 y = βˆ’ 43 20 y = 31 4 12.3xβˆ’2yβˆ’2.5z = 2 36.9x + 7yβˆ’7.5z = βˆ’7 8yβˆ’5z = βˆ’10 0.5xβˆ’3y + 6z = βˆ’0.8 0.7xβˆ’2y = βˆ’0.06 0.5x + 4y + 5z = 0 Extensions For the following exercises, find the inverse of the given matrix. 431. 432. 4331 ⎒ 0 ⎒ 0 ⎣ 1 βˆ’2 ⎒ 1 0 0 2 ⎒ 4 βˆ’2 3 1 ⎣ 1 1 0 βˆ’5 ⎀ βŽ₯ βŽ₯ ⎦ Chapter 11 S...
exercise, if you were told there were 437. 400 more tickets sold for floor 2 than floor 1, how much was the price of each ticket? A food drive collected two different types of canned 438. goods, green beans and kidney beans. The total number of collected cans was 350 and the total weight of all donated food was 348 lb...
solve systems of equations in two variables and three variables, and by multiple methods: substitution, addition, Gaussian elimination, using the inverse of a matrix, and graphing. Some of these methods are easier to apply than others and are more appropriate in certain situations. In this section, we will study two m...
and efficient method for finding solutions to systems with an arbitrary number of unknowns, provided that we have the same number of equations as unknowns. Cramer’s Rule will give us the unique solution to a system of equations, if it exists. However, if the system has no solution or an infinite number of solutions, t...
for y gives (11.4) This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 11 Systems of Equations and Inequalities 1319 a2 b1 y βˆ’ a1 b2 y = a2 c1 βˆ’ a1 c2 y(a2 b1 βˆ’ a1 b2) = a2 c1 βˆ’ a1 c2 y = a2 c1 βˆ’ a1 c2 a2 b1 βˆ’ a1 b2 = a1 c2 βˆ’ a2 c1 a1 b2 βˆ’ a2 b1 a1 c1 a2 c2| = | |a1 b1 a2 b2| Notice that...
ations and Inequalities Solve for x. Solve for y. x = D x 3 D = |15 13 βˆ’3| |12 2 βˆ’3| 3 = βˆ’45 βˆ’ 39 βˆ’36 βˆ’ 6 = βˆ’84 βˆ’42 = 2 y = The solution is (2, βˆ’3). D y D = |12 15 2 13| |12 2 βˆ’3| 3 = 156 βˆ’ 30 βˆ’36 βˆ’ 6 = βˆ’ 126 42 = βˆ’3 11.32 Use Cramer’s Rule to solve the 2 Γ— 2 system of equations. x + 2y = βˆ’11 βˆ’2x + y = βˆ’13 Evaluating t...
cnx.org/content/col11758/1.5 Chapter 11 Systems of Equations and Inequalities 1321 Example 11.59 Finding the Determinant of a 3 Γ— 3 Matrix Find the determinant of the 3 Γ— 3 matrix given A = ⎑ 0 2 1 ⎒ 3 βˆ’1 1 ⎣ 4 0 1 ⎀ βŽ₯ ⎦ Solution Augment the matrix with the first two columns and then follow the formula. Thus, |A| =|0 2...
system using Cramer’s Rule. x + y βˆ’ z = 6 3x βˆ’ 2y + z = βˆ’5 x + 3y βˆ’ 2z = 14 Solution Use Cramer’s Rule. 3 βˆ’2 1 D =|1 1 βˆ’1 1 3 βˆ’2|, D x =| 6 1 βˆ’1 βˆ’5 βˆ’2 1 14 3 βˆ’2|, D y =|1 6 βˆ’1 1 14 βˆ’2|, Dz =|1 1 1 3 14| 6 3 βˆ’2 βˆ’5 3 βˆ’5 1 Then3 βˆ’3 D y D = βˆ’9 βˆ’3 Dz D = 6 βˆ’3 = 1 = 3 = βˆ’ 2 The solution is (1, 3, βˆ’2). 11.34 Use Cramer’s Rul...
y βˆ’ 2z = 0 (2) 2x βˆ’ 4y + 6z = 0 (3) Solution Let’s find the determinant first. Set up a matrix augmented by the first two columns. 1324 Chapter 11 Systems of Equations and Inequalities Then, 3 1 βˆ’2 6 3 2 βˆ’4 |1 βˆ’2 | 1 βˆ’2 1 3 2 βˆ’4| 1(1)(6) + (βˆ’2)(βˆ’2)(2) + 3(3)(βˆ’4) βˆ’ 2(1)(3) βˆ’ (βˆ’4)(βˆ’2)(1) βˆ’ 6(3)(βˆ’2) = 0 As the determinan...
the product of the entries down the main diagonal1 ⎀ βŽ₯ ⎦ Augment A with the first two columns. A = Then ⎑ 1 det(A) = 1(2)(βˆ’1) + 2(1)(0) + 3(0)(0) βˆ’ 0(2)(3) βˆ’ 0(1)(1) + 1(0)(2) = βˆ’2 Property 2 states that interchanging rows changes the sign. Given A = ⎀ ⎑ βˆ’1 5 ⎦, det(A) = (βˆ’1)(βˆ’3) βˆ’ (4)(5) = 3 βˆ’ 20 = βˆ’17 ⎣ 4 βˆ’3 B = ⎑ ⎀...
(2) ⎦, det(B) = 2(4) βˆ’ 3(4) = βˆ’4 ⎣ 3 4 1326 Chapter 11 Systems of Equations and Inequalities Example 11.64 Using Cramer’s Rule and Determinant Properties to Solve a System Find the solution to the given 3 Γ— 3 system. 2x + 4y + 4z = 2 (1) 3x + 7y + 7z = βˆ’5 (2) x + 2y + 2z = 4 (3) Solution Using Cramer’s Rule, we have No...
. 453. 454. 455. 456. 457. 458. 459. 460. 461. 0 2 20 βˆ’1 |1 2 3 4| |βˆ’1 3 βˆ’4| | 2 βˆ’5 6| |βˆ’8 4 βˆ’1 5| |1 3 βˆ’4| |10 0 βˆ’10| |10 0.2 5 0.1| |6 βˆ’3 4| |βˆ’2 3.1 4, 000| |βˆ’1.1 7.2 βˆ’0.5| |βˆ’1 0 0 0 βˆ’3| 0 1 0.6 βˆ’3 0 0 8 463. 462. 464. 465. 0 3 2 5 βˆ’5 βˆ’4 0 2 0 1 0 3 βˆ’4 1 4 1 2 βˆ’8 βˆ’4 βˆ’3 1 |βˆ’1 4 0 0 βˆ’3| |1 0 1 1 0 0| | 2 βˆ’3 1 6 1| |βˆ’2 ...
Rule. x + 2y βˆ’ 4z = βˆ’ 1 7x + 3y + 5z = 26 βˆ’2x βˆ’ 6y + 7z = βˆ’ 6 βˆ’5x + 2y βˆ’ 4z = βˆ’ 47 4x βˆ’ 3y βˆ’ z = βˆ’ 94 3x βˆ’ 3y + 2z = 94 4x + 5y βˆ’ z = βˆ’7 βˆ’2x βˆ’ 9y + 2z = 8 5y + 7z = 21 4x βˆ’ 3y + 4z = 10 5x βˆ’ 2z = βˆ’ 2 3x + 2y βˆ’ 5z = βˆ’ 9 4x βˆ’ 2y + 3z = 6 βˆ’ 6x + y = βˆ’ 2 2x + 7y + 8z = 24 5x + 2y βˆ’ z = 1 βˆ’7x βˆ’ 8y + 3z = 1.5 6x βˆ’ 12y + z =...
the second number, your total is 208 Three numbers add up to 106. The first number is 3 496. less than the second number. The third number is 4 more than the first number. 497. Chapter 11 Systems of Equations and Inequalities 1329 Three numbers add to 216. The sum of the first two numbers is 112. The third number is 8...
scarf cost $11. If you had total revenue of $583, how many yellow scarves and how many purple scarves were sold? Your garden produced two types of tomatoes, one 504. green and one red. The red weigh 10 oz, and the green weigh 4 oz. You have 30 tomatoes, and a total weight of 13 lb, 14 oz. How many of each type of toma...
the 40–49 age group doubled. There are now 6,040 prisoners. Originally, there were 500 more in the 30–39 age group than the 20–29 age group. Determine the prison population for each age group last year. For the following exercises, use this scenario: A healthconscious company decides to make a trail mix out of almonds...
system a system for which there is a single solution to all equations in the system and it is an independent system, or if there are an infinite number of solutions and it is a dependent system cost function costs the function used to calculate the costs of doing business; it usually has two parts, fixed costs and var...
ivalent two matrices A and B are row-equivalent if one can be obtained from the other by performing basic row operations scalar multiple an entry of a matrix that has been multiplied by a scalar solution set the set of all ordered pairs or triples that satisfy all equations in a system of equations substitution method ...
11.5, Example 11.6, and Example 11.7. β€’ Either method of solving a system of equations results in a false statement for inconsistent systems because they are made up of parallel lines that never intersect. See Example 11.8. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 11 Systems o...
(2) one solution, the line is tangent to the parabola; and (3) two solutions, the line intersects the parabola in two points. See Example 11.17. β€’ There are three possible types of solutions to a system of equations representing a circle and a line: (1) no solution, the line does not intersect the circle; (2) one solu...
decomposition of P(x) Q(x), where Q(x) has a repeated irreducible quadratic factor, when the irreducible quadratic factors are repeated, powers of the denominator factors must be represented in increasing powers as Ax + B ⎝ax2 + bx + c⎞ βŽ› ⎠ + A2 x + B2 ⎝ax2 + bx + c⎞ βŽ› ⎠ 2 + β‹― + An x + Bn ⎝ax2 + bx + c⎞ βŽ› ⎠ n. See Exa...
changing rows. β€’ We can use Gaussian elimination to solve a system of equations. See Example 11.39, Example 11.40, and Example 11.41. β€’ Row operations are performed on matrices to obtain row-echelon form. See Example 11.42. β€’ To solve a system of equations, write it in augmented matrix form. Perform row operations to o...
to lower right) and subtract the three diagonal entries (lower left to upper right). See Example 11.59. β€’ To solve a system of three equations in three variables using Cramer’s Rule, replace a variable column with the constant column for each desired solution = Dz D. See Example 11.60. β€’ Cramer’s Rule is also useful f...
) = 150x + 15,000 and revenue R(x) = 200x. What is the break-even point? cost of a a production function 523. A performer charges C(x) = 50x + 10,000, where x is the total number of attendees at a show. The venue charges $75 per ticket. After how many people buy tickets does the venue break even, and what is the value ...
2y βˆ’ z = βˆ’10 x βˆ’ y + 2z = 7 βˆ’x + 3y + z = βˆ’2 This content is available for free at https://cnx.org/content/col11758/1.5 535. y = x2 βˆ’ 4 y = 5x + 10 536. x2 + y2 = 16 y = x βˆ’ 8 537. x2 + y2 = 25 y = x2 + 5 538. x2 + y2 = 4 y βˆ’ x2 = 3 For the following exercises, graph the inequality. 539. y > x2 βˆ’ 1 Chapter 11 Systems ...
)2 Matrices and Matrix Operations the following exercises, perform the requested For operations on the given matrices. A = ⎑ 4 βˆ’2 ⎣ 3 1 ⎀ ⎦, B = ⎑ 6 ⎣ 11 βˆ’2 ⎀ 7 βˆ’3 ⎦, C = 4 ⎑ ⎀ 7 6 ⎒ βŽ₯, D = 11 βˆ’2 ⎣ ⎦ 14 0 ⎑ 1 βˆ’4 ⎒ 10 ⎣ 2 9 5 βˆ’7 5 8 ⎀ βŽ₯, E = ⎦ ⎑ ⎀ 7 βˆ’14 3 ⎒ βŽ₯ 2 βˆ’1 3 ⎣ ⎦ 1 9 0 552. βˆ’4A 566. 567. 568. βˆ’2x + 2y + z = 7 2x ...