Deck 11: Systems of Equations

Question 1

Evaluate A by expanding it along the second row. $$A = \left| \begin{array} { c c c } 4 & 8 & 12 \\16 & 20 & 24 \\28 & 32 & 36\end{array} \right|$$

A) $1$
B) $4$
C) $- 1$
D) $0$
E) $- 2$

Answer 1

$0$

Question 2

Find all solutions of the system. $$\left\{ \begin{aligned}x + y - 3 z + 2 w & = - 1 \\- 2 x - 2 y + 6 z + w & = - 13 \\- x + 3 y + 3 z + 3 w & = - 20 \\2 x + y - 3 z - w & = 10\end{aligned} \right.$$

A) $\left( 2 , - \frac { 3 } { 2 } , - \frac { 3 } { 2 } , - 3 \right)$
B) $\left( - 5 , \frac { 3 } { 2 } , - \frac { 5 } { 2 } , 2 \right)$
C) $\left( - \frac { 3 } { 2 } , - 3 , - 4,2 \right)$
D) $\text { The system is dependent. }$
E) $\text { The system is inconsistent. }$

Answer 2

$\left( 2 , - \frac { 3 } { 2 } , - \frac { 3 } { 2 } , - 3 \right)$

Question 3

Find all solutions of the system. $$\left\{ \begin{array} { r } x + y + z = 12 \\2 x - y - z = - 5 \\6 x + 2 y + z = 30\end{array} \right.$$

A) $\left( \frac { 10 } { 3 } , \frac { 16 } { 3 } , \frac { 10 } { 3 } \right)$
B) $\left( 19 , \frac { 10 } { 3 } , \frac { 7 } { 3 } \right)$
C) $\left( \frac { 7 } { 3 } , \frac { 19 } { 3 } , \frac { 10 } { 3 } \right)$
D) $\text { The system is dependent. }$
E) $\text { The system is inconsistent. }$

Answer 3

$\left( \frac { 7 } { 3 } , \frac { 19 } { 3 } , \frac { 10 } { 3 } \right)$

Question 4

Use a graphing utility to compute the matrix product. $\left( \begin{array} { r r } - 38 & 18 \\23 & 8\end{array} \right)$ $\left( \begin{array} { r r } 89 & - 19 \\17 & 44\end{array} \right)$

A) $\left( \begin{array} { r r } - 3,076 & 1,514 \\2,173 & - 85\end{array} \right)$
B) $\left( \begin{array} { r r } - 3,076 & 1,514 \\2,220 & - 85\end{array} \right)$
C) $\left( \begin{array} { r r } - 3,076 & 1,514 \\2,183 & - 85\end{array} \right)$
D) $\left( \begin{array} { r r } - 3,076 & 1,487 \\2,183 & - 85\end{array} \right)$
E) $\left( \begin{array} { r r } - 3,087 & 1,514 \\2,183 & - 85\end{array} \right)$

Question 5

Use a graphing utility to compute the matrix product. $$\left( \begin{array} { r r r r } - 6 & 9 & - 5 & 1 \\9 & - 1 & - 5 & 2 \\- 5 & - 5 & 9 & - 3 \\1 & 2 & - 3 & 1\end{array} \right) \left( \begin{array} { r r r r } 0.2 & 0.4 & 0.6 & 0.8 \\0.4 & 0.8 & 1.4 & 2.2 \\0.6 & 1.4 & 2.8 & 5 \\0.8 & 2.2 & 5 & 10\end{array} \right)$$

A) $\left( \begin{array} { c c c c } 0.2 & 0 & 0 & 0 \\0 & 0.2 & 0 & 0 \\0 & 0 & 0.2 & 0 \\0 & 0 & 0 & 0.2\end{array} \right)$
B) $\left( \begin{array} { c c c c } 0 & 0 & 0 & 0 \\0.2 & 0.2 & 0 & 0.2 \\0 & 0 & 0 & 0 \\0 & 0 & 0.2 & 0\end{array} \right)$
C) $\left( \begin{array} { c c c c } 0.2 & 0.2 & 0.2 & 0.2 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0\end{array} \right)$
D) $\left( \begin{array} { c c c c } 0 & 0 & 0 & 0 \\0 & 0.2 & 0 & 0.2 \\0.2 & 0 & 0.2 & 0 \\0 & 0 & 0 & 0\end{array} \right)$
E) $\left( \begin{array} { c c c c } 0 & 0 & 0 & 0.2 \\0 & 0 & 0.2 & 0 \\0 & 0.2 & 0 & 0 \\0.2 & 0 & 0 & 0\end{array} \right)$

Question 6

Is $( 5,3 )$ a solution of the following system? $\left\{ \begin{array} { l } 5 x - 3 y = 16 \\2 x + 3 y = 19\end{array} \right.$

Question 7

Find all solutions of the system. $$\left\{ \begin{aligned}- x + y + z + w & = 12 \\2 y - z - w & = 7 \\2 z + 5 w & = 18 \\11 w & = 44\end{aligned} \right.$$

A) $( 4,1 , - 5 , - 4 )$
B) $( - 4,5 , - 1,4 )$
C) $( 5 , - 1 , - 4,4 )$
D) $\text { The system is dependent. }$
E) $\text { The system is inconsistent. }$

Question 8

Find all solutions of the system. $$\left\{ \begin{array} { l } x + y + z + w = 4 \\x - 2 y - z - w = 6 \\2 x - y + z - w = 5 \\x - y + 2 z - 2 w = - 5\end{array} \right.$$

A) $( 0 , - 2,4,5 )$
B) $( 5,0 , - 3,2 )$
C) $( - 8,2 , - 5,0 )$
D) $\text { The system is dependent. }$
E) $\text { The system is inconsistent. }$

Question 9

Use the substitution method to find all solutions of the system. $\left\{ \begin{array} { l } 3 x - 2 y = - 19 \\x + 4 y = - 4\end{array} \right.$

A) $\left( - 6,3 \frac { 1 } { 2 } \right)$
B) $\left( - 6 , - \frac { 1 } { 9 } \right)$
C) $\left( - 6 , - \frac { 1 } { 3 } \right)$
D) $\left( - 6 , \frac { 1 } { 2 } \right)$
E) $\left( - 6 , - \frac { 2 } { 3 } \right)$

Question 10

The matrices A and B are defined as follows: A = $\left( \begin{array} { r r } 4 & 5 \\- 1 & 6\end{array} \right)$ , B = $\left( \begin{array} { r r } 1 & - 1 \\2 & 0\end{array} \right)$ Carry out the indicated matrix operation. $A B$

A) $\left( \begin{array} { c c } 14 & - 4 \\11 & 13\end{array} \right)$
B) $\left( \begin{array} { c c } 14 & - 4 \\11 & 1\end{array} \right)$
C) $\left( \begin{array} { r r } 4 & - 13 \\11 & 1\end{array} \right)$
D) $\left( \begin{array} { c c } 12 & - 4 \\11 & 13\end{array} \right)$
E) $\text { not defined }$

Question 11

The matrices A and B are defined as follows: A = $\left( \begin{array} { r r } 4 & 5 \\- 1 & 6\end{array} \right)$ , B = $\left( \begin{array} { r r } 1 & - 1 \\2 & 0\end{array} \right)$ Carry out the indicated matrix operation.
A + B

A) $\left( \begin{array} { l l } 5 & 4 \\1 & 6\end{array} \right)$
B) $\left( \begin{array} { l l } 1 & 5 \\5 & 6\end{array} \right)$
C) $\left( \begin{array} { l l } 5 & 6 \\1 & 4\end{array} \right)$
D) $\left( \begin{array} { l l } 5 & 4 \\1 & 7\end{array} \right)$
E) $\left( \begin{array} { l l } 5 & 1 \\1 & 5\end{array} \right)$

Question 12

Use matrices to solve the system. $$\left\{ \begin{aligned}5 x + y + 10 z & = 9 \\4 x + 2 y - 10 z & = 84 \\3 x - 4 y & = 11\end{aligned} \right.$$

A) $( 9,9 , - 4 )$
B) $( 9,4 , - 4 )$
C) $( 4,9 , - 4 )$
D) $( 4,4 , - 9 )$
E) $\text { The system is inconsistent. }$

Question 13

Which of the following are linear equations in two variables?

A) $y = x$
B) $\frac { 4 } { x } - \frac { 3 } { y } = - 1$
C) $2 w + 8 z = - 4 w + 3$
D) $y = x ^ { 2 }$
E) $2 x - x = 4$

Question 14

Find all solutions of the system. $\left\{ \begin{array} { r } - x + 6 y + 5 z = 0 \\6 z = 0\end{array} \right.$

A) $( y , 5 y , 6 )$
B) $( 6 y , y , 0 )$
C) $( y , 5 y , 0 )$
D) $\text { The system is dependent. }$
E) $\text { The system is inconsistent. }$

Question 15

Find all solutions of the system. $$\left\{ \begin{array} { l } 2 x - 3 y + 2 z = 3 \\4 x + 2 y + 3 z = 13 \\5 x + 4 y + 2 z = 16\end{array} \right.$$

A) $( 1,2 , - 2 )$
B) $( 1 , - 1,2 )$
C) $( 2,1,1 )$
D) $\text { The system is dependent. }$
E) $\text { The system is inconsistent. }$

Question 16

Solve the system using a graphing utility. $\left\{ \begin{array} { l } 0.05 x - 0.06 y = 1.41 \\0.45 x + 0.60 y = - 0.02\end{array} \right.$

A) $( 17.8 , - 14.7 )$
B) $( 14.8 , - 11.1 )$
C) $( 17.8 , - 11.1 )$
D) $( 14.8 , - 14.7 )$
E) $( 15.8 , - 10.1 )$

Question 17

Evaluate the determinant. $\left| \begin{array} { c c } - 4 & - 9 \\- 7 & 3\end{array} \right|$

A) $75$
B) $- 75$
C) $- 12$
D) $- 63$
E) $- 72$

Question 18

Find the inverse, if it exists. $$A = \left( \begin{array} { c c } \frac { 2 } { 9 } & - \frac { 1 } { 13 } \\- 26 & 9\end{array} \right)$$

A) $A ^ { - 1 } = \left( \begin{array} { c c } - 26 & 13 \\- 5 & 9\end{array} \right)$
B) $A ^ { - 1 } = \left( \begin{array} { c c } 1 & 9 \\2 & - 1\end{array} \right)$
C) $A ^ { - 1 } = \left( \begin{array} { l l } - 1 & 5 \\- 2 & 1\end{array} \right)$
D) $A ^ { - 1 } = \left( \begin{array} { c c } 11 & 6 \\13 & - 8\end{array} \right)$
E) $\text { The inverse does not exist. }$

Question 19

Assume that the supply and demand functions for a commodity are as follows: Supply: q = 200p; Demand: q = 4,200 - 400p.
Find the equilibrium price and the corresponding equilibrium quantity.

A) The equilibrium price is $8 when equilibrium quantity is 1,400 items.
B) The equilibrium price is $7 when equilibrium quantity is 1,300 items.
C) The equilibrium price is $7 when equilibrium quantity is 1,400 items.
D) The equilibrium price is $7 when equilibrium quantity is 1,600 items.
E) The equilibrium price is $6 when equilibrium quantity is 2,400 items.

Question 20

A student in a chemistry laboratory has access to two acid solutions. The first solution is $10 \%$ acid and the second is $35 \%$ acid. (The percentages are by volume.) How many cubic centimeters of each should she mix together to obtain $150 \mathrm {~cm} ^ { 3 }$ of a $25 \%$ acid solution?

A) $60 \mathrm { cc } \text { of } 10 \% \text { solution and } 102 \mathrm { cc } \text { of } 35 \% \text { solution }$
B) $50 \mathrm { cc } \text { of } 10 \% \text { solution and } 90 \mathrm { cc } \text { of } 35 \% \text { solution }$
C) $90 \mathrm { cc } \text { of } 10 \% \text { solution and } 60 \mathrm { cc } \text { of } 35 \% \text { solution }$
D) $60 \mathrm { cc } \text { of } 10 \% \text { solution and } 90 \mathrm { cc } \text { of } 35 \% \text { solution }$
E) $60 \mathrm { cc } \text { of } 10 \% \text { solution and } 100 \mathrm { cc } \text { of } 35 \% \text { solution }$

Question 21

Use Cramer's rule along with a graphing utility to solve the system. $$\left\{ \begin{aligned}2 A - B - 3 C + 2 D = & - 4 \\A - 2 B + C - 3 D = & 8 \\3 A - 4 B + 2 C - 4 D = & 24 \\2 A + 3 B - C - 2 D = & - 8\end{aligned} \right.$$

A) $( 4,2,6,0 )$
B) $( 4 , - 2,6,0 )$
C) $( - 4,2 , - 6 , - 2 )$
D) $( 4 , - 2,6,2 )$
E) $\text { The system is inconsistent. }$

Question 22

Graph the inequality. $$- x - y > 0$$

A)
B)
C)
D)
E)

Question 23

Use Cramer's rule to solve the system. $$\left\{ \begin{aligned}6 x + 4 y - 2 z = & 10 \\2 x - 3 y + 4 z = & 4 \\15 x - 18 z = & - 21\end{aligned} \right.$$

A) $( 2,1,1 )$
B) $( 1,0,2 )$
C) $( 1,2,1 )$
D) $( 1,2,2 )$
E) $\text { The system is inconsistent. }$

Question 24

Graph the inequality. $5 x - 7 y > 35$

A)
B)
C)
D)
E)

Question 25

Use a graphing utility to find all solutions to the nearest one tenth. $$\left\{ \begin{array} { l } y = \sqrt { x + 1 } + 3 \\3 x + 4 y = 12\end{array} \right.$$

A) $( - 0.7,3.5 )$
B) $( - 0.4,3.3 )$
C) $( - 0.6,3.4 )$
D) $( - 1,3.7 )$
E) $( - 0.3,3.2 )$