Question 1

Use Cramer's rule to solve the linear system.
-$$\left\{ \begin{array} { c } 
2 x + 5 y = 21 \
2 x + y = 1
\end{array} \right.$$

Accepted Answer

A)  x = 5, y = -2 
B)  x = -5, y = -2 
C)  x = -2, y = 5 
D)  x = 2, y = -5 
A)  x = 5, y = -2 
B)  x = -5, y = -2 
C)  x = -2, y = 5 
D)  x = 2, y = -5

Question 2

Write the partial fraction decomposition of the rational expression.
-$$\frac { x ^ { 2 } - 56 } { x ^ { 4 } + 5 x ^ { 2 } - 36 }$$

Accepted Answer

The answer of Write the partial fraction decomposition of the...

Question 3

Graph the system of inequalities.
-$$\begin{array} { l } 
2 x + 3 y \geq 6 \
x - y \leq 3 \
x \geq 1
\end{array}$$

A)

Accepted Answer

B)

C) 
  
D) 
  
B)

C) 
  
D)

Question 4

Find the value of the determinant.
-$$\left| \begin{array} { l l } 
6 & 3 \
5 & 2
\end{array} \right|$$

Accepted Answer

A)  -3 
B)  3 
C)  27 
D)  8 
A)  -3 
B)  3 
C)  27 
D)  8

Question 5

Tell whether the given rational expression is proper or improper.
-$$\frac { x ^ { 2 } - 9 x + 20 } { \left( x ^ { 2 } - 11 x + 30 \right) ( x + 5 ) }$$

Accepted Answer

A)  Improper 
B)  Proper 
A)  Improper 
B)  Proper

Question 6

Write the partial fraction decomposition of the rational expression.
-$$\frac { x + 1 } { ( x - 2 ) ^ { 2 } ( x + 4 ) }$$

Accepted Answer

A)  $\frac { 12 } { x - 2 } + \frac { 2 } { ( x - 2 ) ^ { 2 } } + \frac { - 12 } { x + 4 }$ 
B)  $\frac { \frac { 1 } { 2 } } { ( x - 2 ) ^ { 2 } } + \frac { - \frac { 1 } { 12 } } { x + 4 }$ 
C)  $\frac { \frac { 1 } { 12 } } { x - 2 } + \frac { \frac { 1 } { 2 } } { ( x - 2 ) ^ { 2 } } + \frac { - \frac { 1 } { 12 } } { x + 4 }$ 
D)  $\frac { - 1 } { x - 2 } + \frac { \frac { 1 } { 4 } x } { ( x - 2 ) ^ { 2 } } + \frac { - \frac { 1 } { 4 } } { x + 4 }$ 
A)  $\frac { 12 } { x - 2 } + \frac { 2 } { ( x - 2 ) ^ { 2 } } + \frac { - 12 } { x + 4 }$ 
B)  $\frac { \frac { 1 } { 2 } } { ( x - 2 ) ^ { 2 } } + \frac { - \frac { 1 } { 12 } } { x + 4 }$ 
C)  $\frac { \frac { 1 } { 12 } } { x - 2 } + \frac { \frac { 1 } { 2 } } { ( x - 2 ) ^ { 2 } } + \frac { - \frac { 1 } { 12 } } { x + 4 }$ 
D)  $\frac { - 1 } { x - 2 } + \frac { \frac { 1 } { 4 } x } { ( x - 2 ) ^ { 2 } } + \frac { - \frac { 1 } { 4 } } { x + 4 }$

Question 7

Use the properties of determinants to find the value of the second determinant, given the value of the first.
-$$\left| \begin{array} { r r r } 
x & y & z \
u & v & w \
1 & - 1 & 3
\end{array} \right| = - 51 \left| \begin{array} { c c c } 
1 & - 1 & 3 \
2 u & 2 v & 2 w \
x - 1 & y + 1 & z - 3
\end{array} \right| = ?$$

Accepted Answer

A)  -51 
B)  -102 
C)  102 
D)  51 
A)  -51 
B)  -102 
C)  102 
D)  51

Question 8

Solve the system of equations using substitution.
-$$\left\{ \begin{array} { l } 
x y - x ^ { 2 } = - 20 \
x - 2 y = 3
\end{array} \right.$$

Accepted Answer

A)  $x = - 5 , y = - 1 ; x = 8 , y = \frac { 11 } { 2 }$ 
B)  $x = - 5 , y = - 1 ; x = \frac { 11 } { 2 } , y = 8$ 
C)  $x = 5 , y = 1 ; x = - \frac { 11 } { 2 } , y = - 8$ 
D)  $x = 5 , y = 1 ; x = - 8 , y = - \frac { 11 } { 2 }$ 
A)  $x = - 5 , y = - 1 ; x = 8 , y = \frac { 11 } { 2 }$ 
B)  $x = - 5 , y = - 1 ; x = \frac { 11 } { 2 } , y = 8$ 
C)  $x = 5 , y = 1 ; x = - \frac { 11 } { 2 } , y = - 8$ 
D)  $x = 5 , y = 1 ; x = - 8 , y = - \frac { 11 } { 2 }$

Question 9

Solve the problem.
-A coffee store has available 75 pounds of A grade coffee and 120 pounds of B grade coffee. These will be
blended into 1 pound packages as follow: an economy blend that contains 4 ounces of A grade coffee and 12
ounces of B grade coffee and a superior blend that contains 8 ounces of A grade coffee and 8 ounces of B grade
coffee. Using x to denote the number of packages of the economy blend and y to denote the number of packages
of the superior blend, write a system of linear inequalities that describes the possible number of packages of each blend. Graph the system and label the corner points.

Accepted Answer

\[\left\{ \begin{array} { l }

Question 10

Find the inverse of the matrix.
-$$\left[ \begin{array} { l l l } 
1 & 0 & 0 \
5 & 1 & 0 \
0 & 7 & 1
\end{array} \right]$$

Accepted Answer

A)  $\left[ \begin{array} { r r r } 1 & 0 & 0 \ - 5 & 1 & 0 \ 35 & - 7 & 1 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r r } 1 & 7 & - 35 \ 0 & 1 & 1 \ 0 & 0 & 1 \end{array} \right]$ 
C)  $\left[ \begin{array} { l l l } 1 & 0 & 0 \ 5 & 1 & 0 \ 0 & 0 & 7 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r r } 1 & 0 & 0 \ 7 & - 1 & 0 \ - 35 & 5 & 1 \end{array} \right]$ 
A)  $\left[ \begin{array} { r r r } 1 & 0 & 0 \ - 5 & 1 & 0 \ 35 & - 7 & 1 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r r } 1 & 7 & - 35 \ 0 & 1 & 1 \ 0 & 0 & 1 \end{array} \right]$ 
C)  $\left[ \begin{array} { l l l } 1 & 0 & 0 \ 5 & 1 & 0 \ 0 & 0 & 7 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r r } 1 & 0 & 0 \ 7 & - 1 & 0 \ - 35 & 5 & 1 \end{array} \right]$

Question 11

Solve the system.
-$$\left\{ \begin{aligned}
8 x + y & = 2 \
- 24 x - 3 y & = - 6
\end{aligned} \right.$$

Accepted Answer

A)  inconsistent (no solution) 
B)  y = 8x + 6, where x is any real number 
C)  y = -8x + 2, where x is any real number 
D)  x = -8y + 2, where y is any real number 
A)  inconsistent (no solution) 
B)  y = 8x + 6, where x is any real number 
C)  y = -8x + 2, where x is any real number 
D)  x = -8y + 2, where y is any real number

Question 12

Graph the system of inequalities.
-$$\left\{ \begin{aligned}
2 x + 3 y & \geq 6 \
x - y & \leq 3 \
y & \leq 2
\end{aligned} \right.$$

A)

Accepted Answer

B)

C) 
  
D) 
  
B)

C) 
  
D)

Question 13

Solve the problem.
-In a 1-mile race, the winner crosses the finish line 10 feet ahead of the second-place runner and 30 feet ahead of the third-place runner. Assuming that each runner maintains a constant speed throughout the race, by how
Many feet does the second-place runner beat the third-place runner? (5280 feet in 1 mile.)

Accepted Answer

A)  -10.06 ft 
B)  10.02 ft 
C)  20.04 ft 
D)  -20.11 ft 
A)  -10.06 ft 
B)  10.02 ft 
C)  20.04 ft 
D)  -20.11 ft

Question 14

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.
-$$\left\{ \begin{array} { r } 
- 2 x - 8 y = - 8 \
4 x + 16 y = 3
\end{array} \right.$$

Accepted Answer

A)  $x = - 8 , y = 3$ 
B)  $x = - 2 , y = 3$ 
C)  $x = - 2 , y = - 2$ 
D)  Inconsistent 
A)  $x = - 8 , y = 3$ 
B)  $x = - 2 , y = 3$ 
C)  $x = - 2 , y = - 2$ 
D)  Inconsistent

Question 15

Solve the system of equations using elimination.
-$$\left\{ \begin{array} { l } 
x ^ { 2 } + y ^ { 2 } = 64 \
\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 9 } = 1
\end{array} \right.$$

Accepted Answer

A)  x = 0, y= -8; x = 0, y = 8 
B)  x = -8, y= 0; x = 8, y = 0 
C)  x = 0, y= -3; x = 0, y = 3 
D)  inconsistent 
A)  x = 0, y= -8; x = 0, y = 8 
B)  x = -8, y= 0; x = 8, y = 0 
C)  x = 0, y= -3; x = 0, y = 3 
D)  inconsistent

Question 16

Find the inverse of the matrix.
-$\left[ \begin{array} { l l l } 1 & 0 & 8 \ 1 & 2 & 3 \ 2 & 5 & 3 \end{array} \right]$

Accepted Answer

A)  $\left[ \begin{array} { r r r } 9 & - 40 & 16 \ - 3 & 13 & - 5 \ - 1 & 5 & - 2 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r r } - 1 & 0 & - 8 \ - 1 & - 2 & - 3 \ - 2 & - 5 & - 3 \end{array} \right]$ 
C)  $\left[ \begin{array} { l l l } 1 & 1 & 2 \ 0 & 2 & 5 \ 8 & 3 & 3 \end{array} \right]$ 
D)  No inverse 
A)  $\left[ \begin{array} { r r r } 9 & - 40 & 16 \ - 3 & 13 & - 5 \ - 1 & 5 & - 2 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r r } - 1 & 0 & - 8 \ - 1 & - 2 & - 3 \ - 2 & - 5 & - 3 \end{array} \right]$ 
C)  $\left[ \begin{array} { l l l } 1 & 1 & 2 \ 0 & 2 & 5 \ 8 & 3 & 3 \end{array} \right]$ 
D)  No inverse

Question 17

Solve the system of equations using elimination.
-$$\left\{ \begin{array} { l } 
2 x ^ { 2 } - 4 y ^ { 2 } = 2 \
3 x ^ { 2 } + 2 y ^ { 2 } = 35
\end{array} \right.$$

Accepted Answer

A)  x = 3, y = -2; x = 3, y = 2 
B)  x = -3, y = -2; x = -2, y = -3 
C)  x = 3, y = 2; x = -3, y = 2 ; x = 3, y = -2; x = -3, y = -2 
D)  x = 3, y = 2; x = 2, y = 3; x = -3, y = -2; x = -2, y = -3 
A)  x = 3, y = -2; x = 3, y = 2 
B)  x = -3, y = -2; x = -2, y = -3 
C)  x = 3, y = 2; x = -3, y = 2 ; x = 3, y = -2; x = -3, y = -2 
D)  x = 3, y = 2; x = 2, y = 3; x = -3, y = -2; x = -2, y = -3

Question 18

Perform the row operation(s) on the given augmented matrix.
-$$\begin{array} { l } 
\mathrm { R } _ { 1 } = \frac { 1 } { 2 } \mathrm { r } _ { 1 } \
{ \left[ \begin{array} { r r | r } 
2 & - 2 & - 4 \
4 & 2 & - 5
\end{array} \right] }
\end{array}$$

Accepted Answer

A)  $\left[ \begin{array} { r r | r } 1 & - 1 & - 2 \ 2 & 1 & - \frac { 5 } { 2 } \end{array} \right]$ 
B)  $\left[ \begin{array} { r r | r } 1 & - 1 & - 4 \ 4 & 2 & - 5 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r | r } 1 & - 1 & - 2 \ 5 & 1 & - 7 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r | r } 1 & - 1 & - 2 \ 4 & 2 & - 5 \end{array} \right]$ 
A)  $\left[ \begin{array} { r r | r } 1 & - 1 & - 2 \ 2 & 1 & - \frac { 5 } { 2 } \end{array} \right]$ 
B)  $\left[ \begin{array} { r r | r } 1 & - 1 & - 4 \ 4 & 2 & - 5 \end{array} \right]$ 
C)  $\left[ \begin{array} { r r | r } 1 & - 1 & - 2 \ 5 & 1 & - 7 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r | r } 1 & - 1 & - 2 \ 4 & 2 & - 5 \end{array} \right]$

Question 19

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.
-$$\begin{array} { l } 
x + 3 y - 2 z - w = 27 \
4 x + y + z + 2 w = 14 \
- 3 x - y - 3 z - 2 w = - 1 \
x - y - 3 z - 2 w = 11
\end{array}$$

Accepted Answer

A)  $\{ ( 27,14 , - 1,11 ) \}$ 
B)  $\{ ( 3,5 , - 5,1 ) \}$ 
C)  $\{ ( 2,7 , - 9,3 ) \}$ 
D)  $\{ ( 3 + w , 5 - 2 w , - 5 + 2 w , w ) \}$ 
A)  $\{ ( 27,14 , - 1,11 ) \}$ 
B)  $\{ ( 3,5 , - 5,1 ) \}$ 
C)  $\{ ( 2,7 , - 9,3 ) \}$ 
D)  $\{ ( 3 + w , 5 - 2 w , - 5 + 2 w , w ) \}$

Question 20

Perform the indicated matrix operations.
-Let $A = \left[ \begin{array} { r r } - 3 & 3 \ 0 & 2 \end{array} \right]$. Find $3 A$.

Accepted Answer

A)  $\left[ \begin{array} { r r } - 9 & 9 \ 0 & 6 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r } - 9 & 9 \ 0 & 2 \end{array} \right]$ 
C)  $\left[ \begin{array} { l l } 0 & 6 \ 3 & 5 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r } - 9 & 3 \ 0 & 2 \end{array} \right]$ 
A)  $\left[ \begin{array} { r r } - 9 & 9 \ 0 & 6 \end{array} \right]$ 
B)  $\left[ \begin{array} { r r } - 9 & 9 \ 0 & 2 \end{array} \right]$ 
C)  $\left[ \begin{array} { l l } 0 & 6 \ 3 & 5 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r } - 9 & 3 \ 0 & 2 \end{array} \right]$

Use Cramer's rule to solve the linear system. - $\left\{ \begin{array} { c } 2 x + 5 y = 21 \\2 x + y = 1\end{array} \right.$

Write the partial fraction decomposition of the rational expression. - $\frac { x ^ { 2 } - 56 } { x ^ { 4 } + 5 x ^ { 2 } - 36 }$

Find the value of the determinant. - $\left| \begin{array} { l l } 6 & 3 \\5 & 2\end{array} \right|$

Tell whether the given rational expression is proper or improper. - $\frac { x ^ { 2 } - 9 x + 20 } { \left( x ^ { 2 } - 11 x + 30 \right) ( x + 5 ) }$

Write the partial fraction decomposition of the rational expression. - $\frac { x + 1 } { ( x - 2 ) ^ { 2 } ( x + 4 ) }$

Solve the system of equations using substitution. - $\left\{ \begin{array} { l } x y - x ^ { 2 } = - 20 \\x - 2 y = 3\end{array} \right.$

Find the inverse of the matrix. - $\left[ \begin{array} { l l l } 1 & 0 & 0 \\5 & 1 & 0 \\0 & 7 & 1\end{array} \right]$

Solve the system. - $\left\{ \begin{aligned}8 x + y & = 2 \\- 24 x - 3 y & = - 6\end{aligned} \right.$

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent. - $\left\{ \begin{array} { r } - 2 x - 8 y = - 8 \\4 x + 16 y = 3\end{array} \right.$

Solve the system of equations using elimination. - $\left\{ \begin{array} { l } x ^ { 2 } + y ^ { 2 } = 64 \\\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 9 } = 1\end{array} \right.$

Find the inverse of the matrix. - $\left[ \begin{array} { l l l } 1 & 0 & 8 \\ 1 & 2 & 3 \\ 2 & 5 & 3 \end{array} \right]$

Solve the system of equations using elimination. - $\left\{ \begin{array} { l } 2 x ^ { 2 } - 4 y ^ { 2 } = 2 \\3 x ^ { 2 } + 2 y ^ { 2 } = 35\end{array} \right.$

Perform the row operation(s) on the given augmented matrix. - = 2 -2 -4 4 2 -5

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent. - x+3y-2z-w=27 4x+y+z+2w=14 -3x-y-3z-2w=-1 x-y-3z-2w=11

Perform the indicated matrix operations. -Let $A = \left[ \begin{array} { r r } - 3 & 3 \\ 0 & 2 \end{array} \right]$ . Find $3 A$ .

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Exam 10: Systems of Equations and Inequalities

Use Cramer's rule to solve the linear system. - {2x+5y=212x+y=1\left\{ \begin{array} { c } 2 x + 5 y = 21 \\2 x + y = 1\end{array} \right.{2x+5y=212x+y=1​

Write the partial fraction decomposition of the rational expression. - x2−56x4+5x2−36\frac { x ^ { 2 } - 56 } { x ^ { 4 } + 5 x ^ { 2 } - 36 }x4+5x2−36x2−56​

Graph the system of inequalities. - 2x+3y\geq6 x-y\leq3 x\geq1 A) B) C) D)

Find the value of the determinant. - ∣6352∣\left| \begin{array} { l l } 6 & 3 \\5 & 2\end{array} \right|​65​32​​

Tell whether the given rational expression is proper or improper. - x2−9x+20(x2−11x+30)(x+5)\frac { x ^ { 2 } - 9 x + 20 } { \left( x ^ { 2 } - 11 x + 30 \right) ( x + 5 ) }(x2−11x+30)(x+5)x2−9x+20​

Write the partial fraction decomposition of the rational expression. - x+1(x−2)2(x+4)\frac { x + 1 } { ( x - 2 ) ^ { 2 } ( x + 4 ) }(x−2)2(x+4)x+1​

Solve the system of equations using substitution. - {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​

Find the inverse of the matrix. - [100510071]\left[ \begin{array} { l l l } 1 & 0 & 0 \\5 & 1 & 0 \\0 & 7 & 1\end{array} \right]​150​017​001​​

Solve the system. - {8x+y=2−24x−3y=−6\left\{ \begin{aligned}8 x + y & = 2 \\- 24 x - 3 y & = - 6\end{aligned} \right.{8x+y−24x−3y​=2=−6​

Graph the system of inequalities. - {2x+3y≥6x−y≤3y≤2\left\{ \begin{aligned}2 x + 3 y & \geq 6 \\x - y & \leq 3 \\y & \leq 2\end{aligned} \right.⎩⎨⎧​2x+3yx−yy​≥6≤3≤2​ A) B) C) D)

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent. - {−2x−8y=−84x+16y=3\left\{ \begin{array} { r } - 2 x - 8 y = - 8 \\4 x + 16 y = 3\end{array} \right.{−2x−8y=−84x+16y=3​

Solve the system of equations using elimination. - {x2+y2=64x264+y29=1\left\{ \begin{array} { l } x ^ { 2 } + y ^ { 2 } = 64 \\\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 9 } = 1\end{array} \right.{x2+y2=6464x2​+9y2​=1​

Find the inverse of the matrix. - [108123253]\left[ \begin{array} { l l l } 1 & 0 & 8 \\ 1 & 2 & 3 \\ 2 & 5 & 3 \end{array} \right]​112​025​833​​

Solve the system of equations using elimination. - {2x2−4y2=23x2+2y2=35\left\{ \begin{array} { l } 2 x ^ { 2 } - 4 y ^ { 2 } = 2 \\3 x ^ { 2 } + 2 y ^ { 2 } = 35\end{array} \right.{2x2−4y2=23x2+2y2=35​