first two games of the season. Write a matrix containing the total number of points and rebounds for each of the starting five. \[\begin{array}{l} \begin{array} { l | c | c } \text { Game } 1 & \text { Points } & \text { Rebounds } \\ \hline \text { Levy } & 20 & 3 \\ \text { Cowens } & 16 & 5 \\ \text { Williams } & 8 & 12 \\ \text { Miller } & 6 & 11 \\ \text { Jenkins } & 10 & 2 \end{array}\\\\ \begin{array} { l | c | c } \text { Game } 2 & \text { Points } & \text { Rebounds } \\ \hline \text { Levy } & 18 & 4 \\ \text { Cowens } & 14 & 3 \\ \text { Williams } & 12 & 9 \\ \text { Miller } & 4 & 10 \\ \text { Jenkins } & 10 & 3 \end{array} \end{array}\]

A) $\left[ \begin{array} { l l } 5 & 62 \end{array} \right]$ В) $\left[ \begin{array} { r r } 7 & 38 \\ 30 & 8 \\ 21 & 20 \\ 21 & 10 \\ 5 & 20 \end{array} \right]$ C) $\left[ \begin{array} { r r } 38 & 7 \\ 30 & 8 \\ 20 & 21 \\ 10 & 21 \\ 20 & 5 \end{array} \right]$ D) $[ 625 ]$ Answer: C -The table below gives the median annual incomes for men and women in three cities in 2001. Assume that, for all categories, 16% of annual income is withheld to pay taxes. Create a matrix that represents the median annual tax Withholding for men and women in these three cities. Round numbers to the nearest dollar. \[\begin{array} { l | l | l } & \text { Men } & \text { Women } \\ \hline \text { City 1 } & \$ 35,396 & \$ 31,957 \\ \text { City 2 } & 31,544 & 26,940 \\ \text { City 3 } & 26,655 & 24,506 \end{array}\] A) $\left[ \begin{array} { l l } 29,733 & 26,844 \\ 26,497 & 22,630 \\ 22,390 & 20,585 \end{array} \right]$ B) $\left[ \begin{array} { l l } 5113 & 5663 \\ 4310 & 5047 \\ 3921 & 4265 \end{array} \right]$ C) $\left[ \begin{array} { l l } 41,059 & 37,070 \\ 36,591 & 31,250 \\ 30,920 & 28,427 \end{array} \right]$ D) $\left[ \begin{array} { l l } 5663 & 5113 \\ 5047 & 4310 \\ 4265 & 3921 \end{array} \right]$ A) $\left[ \begin{array} { l l } 5 & 62 \end{array} \right]$ В) $\left[ \begin{array} { r r } 7 & 38 \\ 30 & 8 \\ 21 & 20 \\ 21 & 10 \\ 5 & 20 \end{array} \right]$ C) $\left[ \begin{array} { r r } 38 & 7 \\ 30 & 8 \\ 20 & 21 \\ 10 & 21 \\ 20 & 5 \end{array} \right]$ D) $[ 625 ]$ Answer: C -The table below gives the median annual incomes for men and women in three cities in 2001. Assume that, for all categories, 16% of annual income is withheld to pay taxes. Create a matrix that represents the median annual tax Withholding for men and women in these three cities. Round numbers to the nearest dollar. \[\begin{array} { l | l | l } & \text { Men } & \text { Women } \\ \hline \text { City 1 } & \$ 35,396 & \$ 31,957 \\ \text { City 2 } & 31,544 & 26,940 \\ \text { City 3 } & 26,655 & 24,506 \end{array}\] A) $\left[ \begin{array} { l l } 29,733 & 26,844 \\ 26,497 & 22,630 \\ 22,390 & 20,585 \end{array} \right]$ B) $\left[ \begin{array} { l l } 5113 & 5663 \\ 4310 & 5047 \\ 3921 & 4265 \end{array} \right]$ C) $\left[ \begin{array} { l l } 41,059 & 37,070 \\ 36,591 & 31,250 \\ 30,920 & 28,427 \end{array} \right]$ D) $\left[ \begin{array} { l l } 5663 & 5113 \\ 5047 & 4310 \\ 4265 & 3921 \end{array} \right]$

Exam 7: Systems of Equations and Matrices

The following system does not have a unique solution. Solve the system. - $\left\{ \begin{array} { l } x + 4 y - 2 z = - 5 \\3 x - 6 y + 6 z = 7 \\5 x + 2 y + 2 z = - 3\end{array} \right.$

(Multiple Choice)

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Question 81

Find the solution or solutions, if any exist, to the system. - $\left\{ \begin{array} { c } x + y + z = 11 \\x - y + 5 z = 23 \\5 x + y + z = 31\end{array} \right.$

(Multiple Choice)

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Question 82

Either the dimensions of two matrices or the matrices themselves are given. Find the dimensions of the product AB and the product BA. If either is not defined, say so. - $\mathrm { A }$ is $2 \times 1$ ; B is $1 \times 1$ .

(Multiple Choice)

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Question 83

The matrix associated with the solution to a system of linear equations in x, y, and z is given. Write the solution to the system, if it exists. - $\left[ \begin{array} { r r r | r } 1 & 1 & 1 & - 4 \\1 & - 1 & 4 & - 26 \\5 & 1 & 1 & - 24\end{array} \right]$

(Multiple Choice)

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Question 84

$\left\{ \begin{aligned}x - y + 8 z & = - 107 \\6 x + z & = 17 \\3 y - 5 z & = 89\end{aligned} \right.$

(Multiple Choice)

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Question 85

A man has a total of $110,000 invested in three rental properties. One property earns 7.5% per year on its investment, the second earns 9%, and the third earns 10%. The three properties earn a total of $9600 per year. What is the largest amount that could be invested at 7.5%?

(Multiple Choice)

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Question 86

$\left[ \begin{array} { r r r | r } 1 & 0 & 1 & - 2 \\0 & 1 & 2 & - 5 \\0 & 0 & 0 & 0\end{array} \right]$

(Multiple Choice)

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Question 87

Solve using Cramer's rule, if possible. If it is not possible, state whether the system is inconsistent or has infinitely many solutions. - $\left\{ \begin{array} { l } 4 x - 4 y - 4 z = - 20 \\- 8 x + 3 y - 3 z = - 68 \\- 7 x - 7 y - 5 z = - 117\end{array} \right.$

(Multiple Choice)

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Question 88

Either the dimensions of two matrices or the matrices themselves are given. Find the dimensions of the product AB and the product BA. If either is not defined, say so. - $A$ is $2 \times 4$ ; B is $2 \times 4$ .

(Multiple Choice)

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Question 89

Find the indicated matrix. -Let $C = \left[ \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right]$ and $D = \left[ \begin{array} { r } - 1 \\ 3 \\ 2 \end{array} \right]$ . Find $C - 2 D$

(Multiple Choice)

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Question 90

$\left\{ \begin{array} { l } x y - x ^ { 2 } = - 20 \\x - 2 y = 3\end{array} \right.$

(Multiple Choice)

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Question 91

A trust account manager has $500,000 to be invested in three different accounts. The accounts pay 5%, 6%, and 8%, and the goal is to earn $34,000. Assuming that x dollars are invested at 5%, y dollars are invested at 6%, and Z dollars are invested at 8%, find y in terms of z.

(Multiple Choice)

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Question 92

Compute AB, if possible. - $A = \left[ \begin{array} { r r } - 1 & 3 \\5 & 4\end{array} \right] \text { and } B = \left[ \begin{array} { l l l } 0 & - 2 & 6 \\1 & - 3 & 2\end{array} \right]$

(Multiple Choice)

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Question 93

Find the indicated matrix. -Let $C = \left[ \begin{array} { r } 4 \\ - 2 \\ 12 \end{array} \right]$ . Find $\frac { 1 } { 2 } \mathrm { C }$ .

(Multiple Choice)

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Question 94

$\left\{ \begin{aligned}4 x + 2 z + w & = - 18 \\- x + y + 2 z - w & = 4 \\4 x - y + 2 w & = - 21 \\y + 4 z - w & = - 3\end{aligned} \right.$

(Multiple Choice)

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Question 95

A young woman wins $150,000 and buys 5 new cars. She chooses from cars that are priced at $40,000, $30,000, and $25,000 and spends all of the money. How many $25,000 cars did she buy?

(Multiple Choice)

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Question 96

Either the dimensions of two matrices or the matrices themselves are given. Find the dimensions of the product AB and the product BA. If either is not defined, say so. - $\mathrm { A }$ is $3 \times 1$ ; B is $1 \times 3$ .

(Multiple Choice)

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Question 97

A $128,000 trust is to be invested in bonds paying 9%, CDs paying 6%, and mortgages paying 10%. The sum of the bond and CD investment must equal the mortgage investment. To earn an $10,900 annual income from the Investments, how much should the bank invest in bonds?

(Multiple Choice)

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Question 98

Write the augmented matrix associated with the system. - $\left\{ \begin{array} { r } 6 x + 8 z = 64 \\4 y + 4 z = 24 \\6 x + 2 y + 3 z = 62\end{array} \right.$

(Multiple Choice)

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Question 99

first two games of the season. Write a matrix containing the total number of points and rebounds for each of the starting five. Game 1 Points Rebounds Levy 20 3 Cowens 16 5 Williams 8 12 Miller 6 11 Jenkins 10 2 Game 2 Points Rebounds Levy 18 4 Cowens 14 3 Williams 12 9 Miller 4 10 Jenkins 10 3 A) $\left[ \begin{array} { l l } 5 & 62 \end{array} \right]$ В) $\left[ \begin{array} { r r } 7 & 38 \\ 30 & 8 \\ 21 & 20 \\ 21 & 10 \\ 5 & 20 \end{array} \right]$ C) $\left[ \begin{array} { r r } 38 & 7 \\ 30 & 8 \\ 20 & 21 \\ 10 & 21 \\ 20 & 5 \end{array} \right]$ D) $[ 625 ]$ Answer: C -The table below gives the median annual incomes for men and women in three cities in 2001. Assume that, for all categories, 16% of annual income is withheld to pay taxes. Create a matrix that represents the median annual tax Withholding for men and women in these three cities. Round numbers to the nearest dollar. Men Women City 1 \ 35,396 \ 31,957 City 2 31,544 26,940 City 3 26,655 24,506

(Multiple Choice)

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(79)

Question 100

The following system does not have a unique solution. Solve the system. - $\left\{ \begin{array} { l } x + 4 y - 2 z = - 5 \\3 x - 6 y + 6 z = 7 \\5 x + 2 y + 2 z = - 3\end{array} \right.$

Find the solution or solutions, if any exist, to the system. - $\left\{ \begin{array} { c } x + y + z = 11 \\x - y + 5 z = 23 \\5 x + y + z = 31\end{array} \right.$

Either the dimensions of two matrices or the matrices themselves are given. Find the dimensions of the product AB and the product BA. If either is not defined, say so. - $\mathrm { A }$ is $2 \times 1$ ; B is $1 \times 1$ .

The matrix associated with the solution to a system of linear equations in x, y, and z is given. Write the solution to the system, if it exists. - $\left[ \begin{array} { r r r | r } 1 & 1 & 1 & - 4 \\1 & - 1 & 4 & - 26 \\5 & 1 & 1 & - 24\end{array} \right]$

$\left\{ \begin{aligned}x - y + 8 z & = - 107 \\6 x + z & = 17 \\3 y - 5 z & = 89\end{aligned} \right.$

A man has a total of $110,000 invested in three rental properties. One property earns 7.5% per year on its investment, the second earns 9%, and the third earns 10%. The three properties earn a total of $9600 per year. What is the largest amount that could be invested at 7.5%?

$\left[ \begin{array} { r r r | r } 1 & 0 & 1 & - 2 \\0 & 1 & 2 & - 5 \\0 & 0 & 0 & 0\end{array} \right]$

Solve using Cramer's rule, if possible. If it is not possible, state whether the system is inconsistent or has infinitely many solutions. - $\left\{ \begin{array} { l } 4 x - 4 y - 4 z = - 20 \\- 8 x + 3 y - 3 z = - 68 \\- 7 x - 7 y - 5 z = - 117\end{array} \right.$

Either the dimensions of two matrices or the matrices themselves are given. Find the dimensions of the product AB and the product BA. If either is not defined, say so. - $A$ is $2 \times 4$ ; B is $2 \times 4$ .

Find the indicated matrix. -Let $C = \left[ \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right]$ and $D = \left[ \begin{array} { r } - 1 \\ 3 \\ 2 \end{array} \right]$ . Find $C - 2 D$

$\left\{ \begin{array} { l } x y - x ^ { 2 } = - 20 \\x - 2 y = 3\end{array} \right.$

A trust account manager has $500,000 to be invested in three different accounts. The accounts pay 5%, 6%, and 8%, and the goal is to earn $34,000. Assuming that x dollars are invested at 5%, y dollars are invested at 6%, and Z dollars are invested at 8%, find y in terms of z.

Compute AB, if possible. - $A = \left[ \begin{array} { r r } - 1 & 3 \\5 & 4\end{array} \right] \text { and } B = \left[ \begin{array} { l l l } 0 & - 2 & 6 \\1 & - 3 & 2\end{array} \right]$

Find the indicated matrix. -Let $C = \left[ \begin{array} { r } 4 \\ - 2 \\ 12 \end{array} \right]$ . Find $\frac { 1 } { 2 } \mathrm { C }$ .

$\left\{ \begin{aligned}4 x + 2 z + w & = - 18 \\- x + y + 2 z - w & = 4 \\4 x - y + 2 w & = - 21 \\y + 4 z - w & = - 3\end{aligned} \right.$

A young woman wins $150,000 and buys 5 new cars. She chooses from cars that are priced at $40,000, $30,000, and $25,000 and spends all of the money. How many $25,000 cars did she buy?

Either the dimensions of two matrices or the matrices themselves are given. Find the dimensions of the product AB and the product BA. If either is not defined, say so. - $\mathrm { A }$ is $3 \times 1$ ; B is $1 \times 3$ .

A $128,000 trust is to be invested in bonds paying 9%, CDs paying 6%, and mortgages paying 10%. The sum of the bond and CD investment must equal the mortgage investment. To earn an $10,900 annual income from the Investments, how much should the bank invest in bonds?

Write the augmented matrix associated with the system. - $\left\{ \begin{array} { r } 6 x + 8 z = 64 \\4 y + 4 z = 24 \\6 x + 2 y + 3 z = 62\end{array} \right.$

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Exam 7: Systems of Equations and Matrices

The following system does not have a unique solution. Solve the system. - {x+4y−2z=−53x−6y+6z=75x+2y+2z=−3\left\{ \begin{array} { l } x + 4 y - 2 z = - 5 \\3 x - 6 y + 6 z = 7 \\5 x + 2 y + 2 z = - 3\end{array} \right.⎩⎨⎧​x+4y−2z=−53x−6y+6z=75x+2y+2z=−3​

Find the solution or solutions, if any exist, to the system. - {x+y+z=11x−y+5z=235x+y+z=31\left\{ \begin{array} { c } x + y + z = 11 \\x - y + 5 z = 23 \\5 x + y + z = 31\end{array} \right.⎩⎨⎧​x+y+z=11x−y+5z=235x+y+z=31​

Either the dimensions of two matrices or the matrices themselves are given. Find the dimensions of the product AB and the product BA. If either is not defined, say so. - A\mathrm { A }A is 2×12 \times 12×1 ; B is 1×11 \times 11×1 .

{x−y+8z=−1076x+z=173y−5z=89\left\{ \begin{aligned}x - y + 8 z & = - 107 \\6 x + z & = 17 \\3 y - 5 z & = 89\end{aligned} \right.⎩⎨⎧​x−y+8z6x+z3y−5z​=−107=17=89​

A man has a total of $110,000 invested in three rental properties. One property earns 7.5% per year on its investment, the second earns 9%, and the third earns 10%. The three properties earn a total of $9600 per year. What is the largest amount that could be invested at 7.5%?

[101−2012−50000]\left[ \begin{array} { r r r | r } 1 & 0 & 1 & - 2 \\0 & 1 & 2 & - 5 \\0 & 0 & 0 & 0\end{array} \right]​100​010​120​−2−50​​

Either the dimensions of two matrices or the matrices themselves are given. Find the dimensions of the product AB and the product BA. If either is not defined, say so. - AAA is 2×42 \times 42×4 ; B is 2×42 \times 42×4 .

Find the indicated matrix. -Let C=[1−32]C = \left[ \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right]C=​1−32​​ and D=[−132]D = \left[ \begin{array} { r } - 1 \\ 3 \\ 2 \end{array} \right]D=​−132​​ . Find C−2DC - 2 DC−2D

{xy−x2=−20x−2y=3\left\{ \begin{array} { l } x y - x ^ { 2 } = - 20 \\x - 2 y = 3\end{array} \right.{xy−x2=−20x−2y=3​

A trust account manager has $500,000 to be invested in three different accounts. The accounts pay 5%, 6%, and 8%, and the goal is to earn $34,000. Assuming that x dollars are invested at 5%, y dollars are invested at 6%, and Z dollars are invested at 8%, find y in terms of z.

Compute AB, if possible. - A=[−1354] and B=[0−261−32]A = \left[ \begin{array} { r r } - 1 & 3 \\5 & 4\end{array} \right] \text { and } B = \left[ \begin{array} { l l l } 0 & - 2 & 6 \\1 & - 3 & 2\end{array} \right]A=[−15​34​] and B=[01​−2−3​62​]

Find the indicated matrix. -Let C=[4−212]C = \left[ \begin{array} { r } 4 \\ - 2 \\ 12 \end{array} \right]C=​4−212​​ . Find 12C\frac { 1 } { 2 } \mathrm { C }21​C .

{4x+2z+w=−18−x+y+2z−w=44x−y+2w=−21y+4z−w=−3\left\{ \begin{aligned}4 x + 2 z + w & = - 18 \\- x + y + 2 z - w & = 4 \\4 x - y + 2 w & = - 21 \\y + 4 z - w & = - 3\end{aligned} \right.⎩⎨⎧​4x+2z+w−x+y+2z−w4x−y+2wy+4z−w​=−18=4=−21=−3​

A young woman wins $150,000 and buys 5 new cars. She chooses from cars that are priced at $40,000, $30,000, and $25,000 and spends all of the money. How many $25,000 cars did she buy?

Either the dimensions of two matrices or the matrices themselves are given. Find the dimensions of the product AB and the product BA. If either is not defined, say so. - A\mathrm { A }A is 3×13 \times 13×1 ; B is 1×31 \times 31×3 .

A $128,000 trust is to be invested in bonds paying 9%, CDs paying 6%, and mortgages paying 10%. The sum of the bond and CD investment must equal the mortgage investment. To earn an $10,900 annual income from the Investments, how much should the bank invest in bonds?

Write the augmented matrix associated with the system. - {6x+8z=644y+4z=246x+2y+3z=62\left\{ \begin{array} { r } 6 x + 8 z = 64 \\4 y + 4 z = 24 \\6 x + 2 y + 3 z = 62\end{array} \right.⎩⎨⎧​6x+8z=644y+4z=246x+2y+3z=62​

Functions, Graphs, and Models; Linear Functions

Linear Models, Equations, and Inequalities

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Additional Topics With Functions

Exponential and Logarithmic Functions

Higher-Degree Polynomial and Rational Functions

Special Topics in Algebra

Filters

The following system does not have a unique solution. Solve the system. - $\left\{ \begin{array} { l } x + 4 y - 2 z = - 5 \\3 x - 6 y + 6 z = 7 \\5 x + 2 y + 2 z = - 3\end{array} \right.$

Find the solution or solutions, if any exist, to the system. - $\left\{ \begin{array} { c } x + y + z = 11 \\x - y + 5 z = 23 \\5 x + y + z = 31\end{array} \right.$

Either the dimensions of two matrices or the matrices themselves are given. Find the dimensions of the product AB and the product BA. If either is not defined, say so. - $\mathrm { A }$ is $2 \times 1$ ; B is $1 \times 1$ .

$\left\{ \begin{aligned}x - y + 8 z & = - 107 \\6 x + z & = 17 \\3 y - 5 z & = 89\end{aligned} \right.$

$\left[ \begin{array} { r r r | r } 1 & 0 & 1 & - 2 \\0 & 1 & 2 & - 5 \\0 & 0 & 0 & 0\end{array} \right]$

Either the dimensions of two matrices or the matrices themselves are given. Find the dimensions of the product AB and the product BA. If either is not defined, say so. - $A$ is $2 \times 4$ ; B is $2 \times 4$ .

Find the indicated matrix. -Let $C = \left[ \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right]$ and $D = \left[ \begin{array} { r } - 1 \\ 3 \\ 2 \end{array} \right]$ . Find $C - 2 D$

$\left\{ \begin{array} { l } x y - x ^ { 2 } = - 20 \\x - 2 y = 3\end{array} \right.$

Compute AB, if possible. - $A = \left[ \begin{array} { r r } - 1 & 3 \\5 & 4\end{array} \right] \text { and } B = \left[ \begin{array} { l l l } 0 & - 2 & 6 \\1 & - 3 & 2\end{array} \right]$

Find the indicated matrix. -Let $C = \left[ \begin{array} { r } 4 \\ - 2 \\ 12 \end{array} \right]$ . Find $\frac { 1 } { 2 } \mathrm { C }$ .

$\left\{ \begin{aligned}4 x + 2 z + w & = - 18 \\- x + y + 2 z - w & = 4 \\4 x - y + 2 w & = - 21 \\y + 4 z - w & = - 3\end{aligned} \right.$

Either the dimensions of two matrices or the matrices themselves are given. Find the dimensions of the product AB and the product BA. If either is not defined, say so. - $\mathrm { A }$ is $3 \times 1$ ; B is $1 \times 3$ .

Write the augmented matrix associated with the system. - $\left\{ \begin{array} { r } 6 x + 8 z = 64 \\4 y + 4 z = 24 \\6 x + 2 y + 3 z = 62\end{array} \right.$