Question 1

Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants.
$\begin{array} { l } 
a ^ { 2 } x + b ^ { 2 } y = a ^ { 2 } \
a x + b y = b
\end{array}$

Accepted Answer

A)  $\left\{ \left( \frac { a + b } { a } , \frac { b - a } { b } \right) \right\}$ 
B)  $\left\{ \left( \frac { a + b } { a } , - \frac { a } { b } \right) \right\}$ 
C)  $\left\{ \left( \frac { a + b } { a } , \frac { a } { b } \right) \right\}$ 
D)  $\left\{ \left( \frac { a - b } { a } , - \frac { a } { b } \right) \right\}$

Question 2

Solve the system. If the system has infinitely many solutions, write the solution set with x arbitrary.
-$$\begin{array} { l } 
8 x + 4 y = 7 \
16 x + 8 y = 5
\end{array}$$

Accepted Answer

A)  $\varnothing$ 
B)  $\left. \left\{ x , \frac { 7 } { 4 } - 2 x \right) \right\}$ 
C)  $\{ ( x , 1 - 2 x ) \}$ 
D)  $\{ ( 0,0 ) \}$

Question 3

Solve the equation for x.
-$$\left| \begin{array} { l l } 
x & 1 \
2 & x
\end{array} \right| = - 1$$

Accepted Answer

A)  $\{ 1 \}$ 
B)  $\{ - 1 \}$ 
C)  $\{ - 1,1 \}$ 
D)  $\varnothing$

Question 4

A rectangular plot has $126 \mathrm { yd } ^ { 2 }$ with a perimeter of 46 yd. What is the length of the longest side?

Accepted Answer

A)  14 yd 
B)  16 yd 
C)  12 yd 
D)  13 yd

Question 5

Determine the system of inequalities illustrated in the graph. Write inequalities in standard form.
-

Accepted Answer

A)  $y \geq - 2$
$x ^ { 2 } + y \leq 4$ 
B)  $\quad x \geq - 2$
$x ^ { 2 } + y \leq 4$ 
C)  $y \leq - 2$
$x ^ { 2 } + y \geq 4$ 
D)  $\quad y \leq - 2$
$- x ^ { 2 } + y \leq 4$

Question 6

Use the given row transformation to change the matrix as indicated.
-$\left[ \begin{array} { r r r } 3 & 5 & 12 \ 2 & 1 & 2 \ 4 & - 1 & 4 \end{array} \right] ; - 2$ times row 2 added to row 3

Accepted Answer

A) 
$\left[ \begin{array} { r r r } 3 & 5 & 12 \ 2 & 1 & 2 \ 8 & 1 & 8 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r r } 3 & 5 & 12 \ 2 & 1 & 2 \ 0 & - 2 & 2 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r r } 3 & 5 & 12 \ 2 & 1 & 2 \ 0 & - 3 & 0 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r r } - 1 & 3 & 8 \ 2 & 1 & 2 \ 4 & - 1 & 4 \end{array} \right]$

Question 7

Find the matrix product when possible.
-$$\left[ \begin{array} { r r r } 
1 & 3 & - 3 \
3 & 0 & 5
\end{array} \right] \left[ \begin{array} { r r } 
3 & 0 \
- 3 & 1 \
0 & 5
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { r r } 
- 12 & - 6 \
25 & - 9
\end{array} \right]$$ 
B)  They cannot be multiplied. 
C) 
$$\left[ \begin{array} { r r r } 
3 & 3 & 0 \
0 & 0 & 25
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { r r } 
- 6 & - 12 \
9 & 25
\end{array} \right]$$

Question 8

Find the partial fraction decomposition for the rational expression.
$\frac { 8 x ^ { 2 } - 97 x + 18 } { \left( x ^ { 2 } + 3 x - 6 \right) ( x - 8 ) }$

Accepted Answer

A)  $\frac { 11 x } { x ^ { 2 } + 3 x - 6 } + \frac { - 3 } { x - 8 }$ 
B)  $\frac { 11 x } { x ^ { 2 } + 3 x - 6 } + \frac { 3 } { x - 8 }$ 
C)  $\frac { 11 x } { x - 8 } + \frac { - 3 } { x ^ { 2 } + 3 x - 6 }$ 
D)  $\frac { 3 x } { x ^ { 2 } + 3 x - 6 } + \frac { - 11 } { x - 8 }$

Question 9

Provide an appropriate response.
-Indicate whether the following statement is If two of the rows of a 3 × 3 matrix are identical, but its
other row is different, then the determinant of the matrix equals 0.

Accepted Answer

A) True 
 B)False

Question 10

Solve the problem using matrices.
-Ellen wishes to mix candy worth $2.19 per pound with candy worth $7.87 per pound to form 22 pounds of a mixture worth $6.58 per pound. How many pounds of the more expensive candy should she use?

Accepted Answer

A)  5 lb 
B)  17 lb 
C)  7 lb 
D)  22 lb

Question 11

If the system has infinitely many solutions, write the solution set with x arbitrary.
-$$\begin{array} { r } 
4 x + y + z = 2 \
x - y + 5 z = 8 \
4 x + y + z = 11
\end{array}$$

Accepted Answer

A)  $\{ ( 2,3 , - 3 ) \}$ 
B)  $\varnothing$ 
C)  $\{ ( 2 , - 3,3 ) \}$ 
D)  $\{ ( 3 , - 3,2 ) \}$

Question 12

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, let the last variable be the arbitrary variable.
$\begin{array} { l } 
2 x - 2 y + 3 z = - 3 \
x + 2 y - z = 9 \
2 y + z = - 1
\end{array}$

Accepted Answer

A)  $\{ ( 4,1 , - 3 ) \}$ 
B)  $\varnothing$ 
C)  $\{ ( 6,8 , - 7 ) \}$ 
D)  $\{ ( - 4,1,13 ) \}$

Question 13

Use the given row transformation to change the matrix as indicated.
-$\left[ \begin{array} { r r r } 2 & 1 & 2 \ - 4 & 3 & - 1 \ 7 & 4 & 0 \end{array} \right] ; 2$ times row 1 added to row 2

Accepted Answer

A) 
$\left[ \begin{array} { l l l } 4 & 2 & 4 \ 0 & 5 & 3 \ 7 & 4 & 0 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r r } 2 & 1 & 2 \ - 4 & 3 & - 1 \ 11 & 6 & 4 \end{array} \right]$ 
C) 
$\left[\begin{array}{lll}
2 & 1 & 2 \
0 & 5 & 3 \
7 & 4 & 0
\end{array}\right]$ 
D) 
$\left[\begin{array}{lll}
2 & 1 & 2 \
0 & 3 & 3 \
7 & 4 & 0
\end{array}\right]$

Question 14

Provide an appropriate response.
-Which one of the following is a description of the graph of the inequality? $$( x + 6 ) ^ { 2 } + ( y - 3 ) ^ { 2 } < 81$$

Accepted Answer

A)  The region inside a circle with center $( - 6,3 )$ and radius 9 
B)  The region outside a circle with center $( 6 , - 3 )$ and radius 9 
C)  The region inside the circle with center $( 6 , - 3 )$ and radius 9 
D)  The region outside a circle with center $( - 6,3 )$ and radius 9

Question 15

Solve the problem using matrices.
-Jerry has three pieces of string. The total length is 107 cm. The sum of the two longer pieces is 89 cm. The middle piece is twice as long as the shortest piece. What are the lengths of the three pieces?

Accepted Answer

A)  No solution 
B)  18 cm; 36 cm; 53 cm 
C)  9 cm; 18 cm; 80 cm 
D)  18 cm; 53 cm; 63 cm

Question 16

Le Boulangerie, a bakery, sells four main items: sweet rolls, bread, cakes, and pies. The amount of each ingredient (in cups, except for eggs) required for these items is given by matrix A. 
Suppose a day's orders consist of 20 dozen sweet rolls, 200 loaves of bread, 50 cakes, and 60 pies. Write the orders as a $1 \times 4$ matrix and, using matrix multiplication, write as a matrix the amount of each ingredient needed to fill the day's orders.

Accepted Answer

A)  $[ 66075413512570 ]$ 
B)  $\left[ \begin{array} { l l l l l } 660 & 1150 & 150 & 125 & 250 \end{array} \right]$ 
C)  
D)  These matrices cannot be multiplied.

Question 17

Carole's car averages 17.4 mi/gal in city driving and 25.7 mi/gal in highway driving. If she drove a total of 523.9 mi on 22 gal of gas, how many of the gallons were used for city driving?

Accepted Answer

A)  5 gal 
B)  7 gal 
C)  17 gal 
D)  22 gal

Question 18

Use the shading capabilities of your graphing calculator to graph the inequality or system of inequalities.
-$y \geq x^{2}-3$

Accepted Answer

A) 
 
B) 
 
C) 
 
D)

Question 19

Solve the system.
$\begin{array} { l } 
4 x + 5 y + z = - 33 \
5 x - 3 y - z = - 9 \
3 x + y + 4 z = - 23
\end{array}$

Accepted Answer

A)  $\{ ( - 4 , - 3 , - 2 ) \}$ 
B)  $\{ ( - 2 , - 3 , - 4 ) \}$ 
C)  $\{ ( - 4 , - 2 , - 3 ) \}$ 
D)  $\varnothing$

Question 20

Give all solutions of the nonlinear system of equations, including those with nonreal complex components.
$\begin{array} { l } 
x ^ { 2 } - y ^ { 2 } = 39 \
x - y = 3
\end{array}$

Accepted Answer

A)  {(-8,5)} 
B)  {(8,-5)} 
C)  {(8,5)} 
D)  {(-8,-5)}

Exam 6: Systems and Matrices

Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants. $\begin{array} { l } a ^ { 2 } x + b ^ { 2 } y = a ^ { 2 } \\a x + b y = b\end{array}$

Solve the system. If the system has infinitely many solutions, write the solution set with x arbitrary. - $\begin{array} { l } 8 x + 4 y = 7 \\16 x + 8 y = 5\end{array}$

Solve the equation for x. - $\left| \begin{array} { l l } x & 1 \\2 & x\end{array} \right| = - 1$

A rectangular plot has $126 \mathrm { yd } ^ { 2 }$ with a perimeter of 46 yd. What is the length of the longest side?

Use the given row transformation to change the matrix as indicated. - $\left[ \begin{array} { r r r } 3 & 5 & 12 \\ 2 & 1 & 2 \\ 4 & - 1 & 4 \end{array} \right] ; - 2$ times row 2 added to row 3

Find the matrix product when possible. - $\left[ \begin{array} { r r r } 1 & 3 & - 3 \\3 & 0 & 5\end{array} \right] \left[ \begin{array} { r r } 3 & 0 \\- 3 & 1 \\0 & 5\end{array} \right]$

Find the partial fraction decomposition for the rational expression. $\frac { 8 x ^ { 2 } - 97 x + 18 } { \left( x ^ { 2 } + 3 x - 6 \right) ( x - 8 ) }$

Provide an appropriate response. -Indicate whether the following statement is If two of the rows of a 3 × 3 matrix are identical, but its other row is different, then the determinant of the matrix equals 0.

Solve the problem using matrices. -Ellen wishes to mix candy worth $2.19 per pound with candy worth $7.87 per pound to form 22 pounds of a mixture worth $6.58 per pound. How many pounds of the more expensive candy should she use?

If the system has infinitely many solutions, write the solution set with x arbitrary. - $\begin{array} { r } 4 x + y + z = 2 \\x - y + 5 z = 8 \\4 x + y + z = 11\end{array}$

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, let the last variable be the arbitrary variable. $\begin{array} { l } 2 x - 2 y + 3 z = - 3 \\x + 2 y - z = 9 \\2 y + z = - 1\end{array}$

Use the given row transformation to change the matrix as indicated. - $\left[ \begin{array} { r r r } 2 & 1 & 2 \\ - 4 & 3 & - 1 \\ 7 & 4 & 0 \end{array} \right] ; 2$ times row 1 added to row 2

Provide an appropriate response. -Which one of the following is a description of the graph of the inequality? $( x + 6 ) ^ { 2 } + ( y - 3 ) ^ { 2 } < 81$

Solve the problem using matrices. -Jerry has three pieces of string. The total length is 107 cm. The sum of the two longer pieces is 89 cm. The middle piece is twice as long as the shortest piece. What are the lengths of the three pieces?

Carole's car averages 17.4 mi/gal in city driving and 25.7 mi/gal in highway driving. If she drove a total of 523.9 mi on 22 gal of gas, how many of the gallons were used for city driving?

Use the shading capabilities of your graphing calculator to graph the inequality or system of inequalities. - $y \geq x^{2}-3$ $Use the shading capabilities of your graphing calculator to graph the inequality or system of inequalities. - y \geq x^{2}-3 A) B) C) D)$

Solve the system. $\begin{array} { l } 4 x + 5 y + z = - 33 \\5 x - 3 y - z = - 9 \\3 x + y + 4 z = - 23\end{array}$

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. $\begin{array} { l } x ^ { 2 } - y ^ { 2 } = 39 \\x - y = 3\end{array}$

Exam 6: Systems and Matrices

Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants. a2x+b2y=a2ax+by=b\begin{array} { l } a ^ { 2 } x + b ^ { 2 } y = a ^ { 2 } \\a x + b y = b\end{array}a2x+b2y=a2ax+by=b​

Solve the system. If the system has infinitely many solutions, write the solution set with x arbitrary. - 8x+4y=716x+8y=5\begin{array} { l } 8 x + 4 y = 7 \\16 x + 8 y = 5\end{array}8x+4y=716x+8y=5​

Solve the equation for x. - ∣x12x∣=−1\left| \begin{array} { l l } x & 1 \\2 & x\end{array} \right| = - 1​x2​1x​​=−1

A rectangular plot has 126yd2126 \mathrm { yd } ^ { 2 }126yd2 with a perimeter of 46 yd. What is the length of the longest side?

Determine the system of inequalities illustrated in the graph. Write inequalities in standard form. -

Use the given row transformation to change the matrix as indicated. - [35122124−14];−2\left[ \begin{array} { r r r } 3 & 5 & 12 \\ 2 & 1 & 2 \\ 4 & - 1 & 4 \end{array} \right] ; - 2​324​51−1​1224​​;−2 times row 2 added to row 3

Find the matrix product when possible. - [13−3305][30−3105]\left[ \begin{array} { r r r } 1 & 3 & - 3 \\3 & 0 & 5\end{array} \right] \left[ \begin{array} { r r } 3 & 0 \\- 3 & 1 \\0 & 5\end{array} \right][13​30​−35​]​3−30​015​​

Find the partial fraction decomposition for the rational expression. 8x2−97x+18(x2+3x−6) (x−8) \frac { 8 x ^ { 2 } - 97 x + 18 } { \left( x ^ { 2 } + 3 x - 6 \right) ( x - 8 ) }(x2+3x−6) (x−8) 8x2−97x+18​

Provide an appropriate response. -Indicate whether the following statement is If two of the rows of a 3 × 3 matrix are identical, but its other row is different, then the determinant of the matrix equals 0.

Solve the problem using matrices. -Ellen wishes to mix candy worth $2.19 per pound with candy worth $7.87 per pound to form 22 pounds of a mixture worth $6.58 per pound. How many pounds of the more expensive candy should she use?

If the system has infinitely many solutions, write the solution set with x arbitrary. - 4x+y+z=2x−y+5z=84x+y+z=11\begin{array} { r } 4 x + y + z = 2 \\x - y + 5 z = 8 \\4 x + y + z = 11\end{array}4x+y+z=2x−y+5z=84x+y+z=11​

Use the given row transformation to change the matrix as indicated. - [212−43−1740];2\left[ \begin{array} { r r r } 2 & 1 & 2 \\ - 4 & 3 & - 1 \\ 7 & 4 & 0 \end{array} \right] ; 2​2−47​134​2−10​​;2 times row 1 added to row 2

Provide an appropriate response. -Which one of the following is a description of the graph of the inequality? (x+6) 2+(y−3) 2<81( x + 6 ) ^ { 2 } + ( y - 3 ) ^ { 2 } < 81(x+6) 2+(y−3) 2<81

Solve the problem using matrices. -Jerry has three pieces of string. The total length is 107 cm. The sum of the two longer pieces is 89 cm. The middle piece is twice as long as the shortest piece. What are the lengths of the three pieces?

Carole's car averages 17.4 mi/gal in city driving and 25.7 mi/gal in highway driving. If she drove a total of 523.9 mi on 22 gal of gas, how many of the gallons were used for city driving?

Use the shading capabilities of your graphing calculator to graph the inequality or system of inequalities. - y≥x2−3y \geq x^{2}-3y≥x2−3

Solve the system. 4x+5y+z=−335x−3y−z=−93x+y+4z=−23\begin{array} { l } 4 x + 5 y + z = - 33 \\5 x - 3 y - z = - 9 \\3 x + y + 4 z = - 23\end{array}4x+5y+z=−335x−3y−z=−93x+y+4z=−23​

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. x2−y2=39x−y=3\begin{array} { l } x ^ { 2 } - y ^ { 2 } = 39 \\x - y = 3\end{array}x2−y2=39x−y=3​

Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants. $\begin{array} { l } a ^ { 2 } x + b ^ { 2 } y = a ^ { 2 } \\a x + b y = b\end{array}$

Solve the system. If the system has infinitely many solutions, write the solution set with x arbitrary. - $\begin{array} { l } 8 x + 4 y = 7 \\16 x + 8 y = 5\end{array}$

Solve the equation for x. - $\left| \begin{array} { l l } x & 1 \\2 & x\end{array} \right| = - 1$

A rectangular plot has $126 \mathrm { yd } ^ { 2 }$ with a perimeter of 46 yd. What is the length of the longest side?

Use the given row transformation to change the matrix as indicated. - $\left[ \begin{array} { r r r } 3 & 5 & 12 \\ 2 & 1 & 2 \\ 4 & - 1 & 4 \end{array} \right] ; - 2$ times row 2 added to row 3

Find the matrix product when possible. - $\left[ \begin{array} { r r r } 1 & 3 & - 3 \\3 & 0 & 5\end{array} \right] \left[ \begin{array} { r r } 3 & 0 \\- 3 & 1 \\0 & 5\end{array} \right]$

Find the partial fraction decomposition for the rational expression. $\frac { 8 x ^ { 2 } - 97 x + 18 } { \left( x ^ { 2 } + 3 x - 6 \right) ( x - 8 ) }$

If the system has infinitely many solutions, write the solution set with x arbitrary. - $\begin{array} { r } 4 x + y + z = 2 \\x - y + 5 z = 8 \\4 x + y + z = 11\end{array}$

Use the given row transformation to change the matrix as indicated. - $\left[ \begin{array} { r r r } 2 & 1 & 2 \\ - 4 & 3 & - 1 \\ 7 & 4 & 0 \end{array} \right] ; 2$ times row 1 added to row 2

Provide an appropriate response. -Which one of the following is a description of the graph of the inequality? $( x + 6 ) ^ { 2 } + ( y - 3 ) ^ { 2 } < 81$

Use the shading capabilities of your graphing calculator to graph the inequality or system of inequalities. - $y \geq x^{2}-3$ $Use the shading capabilities of your graphing calculator to graph the inequality or system of inequalities. - y \geq x^{2}-3 A) B) C) D)$

Solve the system. $\begin{array} { l } 4 x + 5 y + z = - 33 \\5 x - 3 y - z = - 9 \\3 x + y + 4 z = - 23\end{array}$

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. $\begin{array} { l } x ^ { 2 } - y ^ { 2 } = 39 \\x - y = 3\end{array}$