Question 1

Determine which one of the ordered triples below is a solution of the given system of equations.
$$\left\{ \begin{array} { l } 
8 x + 6 y - 4 z = 74 \
x - 5 y + 8 z = - 102 \
5 x - 3 y + 2 z = - 46
\end{array} \right.$$

Accepted Answer

A)  $( 4,7,0 )$ 
B)  $( - 1,9 , - 7 )$ 
C)  $( 9,7,1 )$ 
D)  $( 4,7 , - 7 )$ 
E)  $( - 7,9 , - 1 )$

Question 2

An object moving vertically is at the given heights at the specified times. Find the position equation $s = \frac { 1 } { 2 } a t ^ { 2 } + v _ { o } t + s _ { o }$ for the object.
At $t = 1$ second, $s = 198$ feet
At $t = 2$ seconds, $s = 182$ feet
At $t = 3$ seconds, $s = 134$ feet

Accepted Answer

A)  $s = - 32 t ^ { 2 } + 32 t + 198$ 
B)  $s = - 16 t ^ { 2 } + 32 t - 182$ 
C)  $s = - 16 t ^ { 2 } + 32 t + 182$ 
D)  $s = - 8 t ^ { 2 } + 2 t - 182$ 
E)  $\quad s = - 16 t ^ { 2 } - 32 t + 198$

Question 3

Write the augmented matrix for the system of linear equations.
$$\left\{ \begin{aligned}
x + 4 y - z & = - 9 \
- 9 y + 4 z & = - 6 \
x + 5 z & = - 4
\end{aligned} \right.$$

Accepted Answer

A)  $\left[ \begin{array} { c c c : c } 1 & 4 & - 1 & - 9 \ 1 & - 9 & 4 & - 6 \ 1 & 1 & 5 & - 4 \end{array} \right]$ 
B) 
$\left[ \begin{array} { c c c : c } 1 & 4 & - 1 & - 9 \ 0 & - 9 & 4 & - 6 \ 1 & 0 & 5 & - 4 \end{array} \right]$ 
C) 
$\left[ \begin{array} { c c c : c } 1 & 4 & - 1 & - 9 \ & - 9 & 4 & - 6 \ 1 & & 5 & - 4 \end{array} \right]$ 
D) 
$\left[ \begin{array} { c c c : c } 1 & 4 & - 1 & - 9 \ - 9 & 4 & 0 & - 6 \ 1 & 5 & 0 & - 4 \end{array} \right]$ 
E) 
$\left[ \begin{array} { c c c : c } 1 & 4 & - 1 & - 9 \ 0 & - 9 & 4 & - 6 \ 1 & 5 & 0 & - 4 \end{array} \right]$ ]

Question 4

Solve using any method.
$$\left\{ \begin{aligned}
7 x + 6 y & = 15 \
y & = x - 7
\end{aligned} \right.$$

Accepted Answer

A)  $( 7,0 )$ 
B)  $\left( - 6 , \frac { 51 } { 7 } \right)$ 
C)  $\left( - \frac { 27 } { 13 } , - \frac { 64 } { 13 } \right)$ 
D)  $\left( \frac { 57 } { 13 } , - \frac { 34 } { 13 } \right)$ 
E)  inconsistent

Question 5

Use a determinant to determine whether the points below are collinear.
$$( 3 , - 1 ) , ( 0 , - 3 ) , ( 24,13 )$$

Accepted Answer

A)  the points are collinear 
B)  the points are not collinear

Question 6

Solve the system of linear equations.
$$\left\{ \begin{aligned}
2 x - 2 y + 3 z & = - 21 \
2 x + z & = - 11 \
- 2 x + 4 y - 4 z & = 28
\end{aligned} \right.$$

Accepted Answer

A)  $( 3 , - 3,4 )$ 
B)  $( - 4,2 , - 3 )$ 
C)  $( 2 , - 3 , - 4 )$ 
D)  $( - 3,4,3 )$ 
E)  $( - 4,2,6 )$

Question 7

An augmented matrix that represents a system of linear equations (in variables $x , y$, and $z$ ) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix.
$$\left[ \begin{array} { c c c : c } 
1 & 0 & 0 & 9 \
0 & 1 & 0 & 8 \
0 & 0 & 1 & - 7
\end{array} \right]$$

Accepted Answer

A)  $x = 9 , y = 8 , z = - 7$ 
B)  $x = - 9 , y = - 8 , z = 7$ 
C)  $x = 0 , y = 0 , z = 0$ 
D)  $x = 9 x , y = - 7 y , z = 8 z$ 
E)  $x = 9 , y = 0 , z = 0$

Question 8

Write the partial fraction decomposition of the improper rational expression.
$$\frac { x ^ { 3 } - 4 x ^ { 2 } + 3 x + 2 } { x ^ { 2 } + x - 2 }$$

Accepted Answer

A)  $x - 5 + \frac { 1 } { 3 } \left( \frac { 2 } { x - 1 } + \frac { 28 } { x + 2 } \right)$ 
B)  $x - 5 + \frac { 1 } { 3 } \left( \frac { 18 } { x - 1 } + \frac { 28 } { x + 2 } \right)$ 
C)  $x - 5 + \frac { 1 } { 3 } \left( \frac { 18 } { x + 1 } + \frac { 28 } { x - 2 } \right)$ 
D)  $x - 5 + \frac { 1 } { 3 } \left( \frac { 2 } { x - 1 } - \frac { 28 } { x + 2 } \right)$ 
E)  $\frac { 1 } { 3 } \left( \frac { 2 } { x - 1 } + \frac { 28 } { x + 2 } \right)$

Question 9

Solve the system by the method of elimination.
$$\left\{ \begin{array} { c } 
5 x - y = 11 \
- 7 x - y = - 25
\end{array} \right.$$

Accepted Answer

A)  $( 1,18 )$ 
B)  $( 3,4 )$ 
C)  $\left( - \frac { 7 } { 6 } , - \frac { 157 } { 6 } \right)$ 
D)  $\left( - 5 , \frac { 6 } { 5 } \right)$ 
E)  inconsistent

Question 10

Find the minor $M _ { 13 }$ and its cofactor $C _ { 13 }$ of the matrix $\left[ \begin{array} { c c c } - 9 & 6 & - 24 \ 9 & - 6 & 18 \ - 3 & 9 & - 18 \end{array} \right]$.

Accepted Answer

A) 
$M _ { 13 } = \left| \begin{array} { c c } - 9 & - 24 \ - 3 & - 18 \end{array} \right| = 90$
$C _ { 13 } = - 90$ 
B) 
$M _ { 13 } = \left| \begin{array} { c c } 6 & - 24 \ - 6 & 18 \end{array} \right| = - 36$
$C _ { 13 } = - 36$ 
C) 
$M _ { 13 } = \left| \begin{array} { c c } 9 & - 6 \ - 3 & 9 \end{array} \right| = 63$
$C _ { 13 } = - 63$ 
D) 
$M _ { 13 } = \left| \begin{array} { c c } 6 & - 24 \ - 6 & 18 \end{array} \right| = - 36$
$C _ { 13 } = 36$ 
E) 
$M _ { 13 } = \left| \begin{array} { c c } 9 & - 6 \ - 3 & 9 \end{array} \right| = 63$
$C _ { 13 } = 63$

Question 11

Solve the system of linear equations.
$$\left\{ \begin{aligned}
x + y + z + w & = 11 \
- 2 x - y - 3 z + 4 w & = - 5 \
- 2 x - 5 z - w & = - 23 \
3 x + 5 y + z - 2 w & = 16
\end{aligned} \right.$$

Accepted Answer

A)  $( 5,1,3,2 )$ 
B)  $( 5,2,1,3 )$ 
C)  $( 3,1,2,5 )$ 
D)  $( 5,1,2,3 )$ 
E)  inconsistent

Question 12

Solve the system by the method of elimination. Round numbers to three decimal places.
$$\left\{ \begin{array} { c } 
5.7 x + 7.1 y = 3.2 \
- 0.8 x - 4.8 y = - 1.1
\end{array} \right.$$

Accepted Answer

A)  $( 2.872,5.844 )$ 
B)  $( 0.348,0.171 )$ 
C)  $( 0.229,0.267 )$ 
D)  $( 0.175,0.310 )$ 
E)  no solution

Question 13

Evaluate the determinant $\left| \begin{array} { c c } 7 x & x \ln x \ 7 & 2 + \ln x \end{array} \right|$ in which the entries are functions.

Accepted Answer

A)  $49 \ln x$ 
B)  $14 x$ 
C)  $2 x$ 
D)  $7 - \ln x$ 
E)  $2 \ln x$

Question 14

Find $| A B |$, if
$$A = \left[ \begin{array} { r r r } 
5 & - 2 & - 2 \
- 3 & 4 & - 1 \
- 5 & 6 & 2
\end{array} \right] , B = \left[ \begin{array} { r r r } 
1 & - 1 & - 3 \
- 5 & - 3 & 4 \
- 5 & - 1 & - 2
\end{array} \right]$$

Accepted Answer

A)  3080 
B)  616 
C)  150 
D)  280 
E)  1300

Question 15

Solve the system of linear equations
$\left\{ \begin{array} { l } 
4 x _ { 1 } - 8 x _ { 2 } - 4 x _ { 3 } - 8 x _ { 4 } = 0 \
12 x _ { 1 } - 20 x _ { 2 } - 8 x _ { 3 } - 12 x _ { 4 } = - 9 \
8 x _ { 1 } - 20 x _ { 2 } - 8 x _ { 3 } - 20 x _ { 4 } = 6 \
- 4 x _ { 1 } + 16 x _ { 2 } + 16 x _ { 3 } + 44 x _ { 4 } = 0
\end{array} \right.$ 
using the inverse matrix $\frac { 1 } { 4 } \left[ \begin{array} { c c c c } - 24 & 7 & 1 & - 2 \ - 10 & 3 & 0 & - 1 \ - 29 & 7 & 3 & - 2 \ 12 & - 3 & - 1 & 1 \end{array} \right]$.

Accepted Answer

A)  
$\left[\begin{array}{l}
x_{1} \
x_{2} \
x_{3} \
x_{4}
\end{array}\right]=\left[\begin{array}{r}
-\frac{57}{4} \
-\frac{27}{4} \
-\frac{45}{4} \
\frac{21}{4}
\end{array}\right]$ 
B)  $\left[ \begin{array} { l } x _ { 1 } \ x _ { 2 } \ x _ { 3 } \ x _ { 4 } \end{array} \right] = \left[ \begin{array} { c } - \frac { 21 } { 4 } \ - \frac { 15 } { 4 } \ \frac { 27 } { 4 } \ - \frac { 27 } { 4 } \end{array} \right]$ 
C)  $\left[ \begin{array} { l } x _ { 1 } \ x _ { 2 } \ x _ { 3 } \ x _ { 4 } \end{array} \right] = \left[ \begin{array} { c } - \frac { 27 } { 4 } \ - \frac { 57 } { 4 } \ 0 \ \frac { 21 } { 4 } \end{array} \right]$ 
D) 
$\left[ \begin{array} { l } x _ { 1 } \ x _ { 2 } \ x _ { 3 } \ x _ { 4 } \end{array} \right] = \left[ \begin{array} { c } - \frac { 9 } { 4 } \ \frac { 3 } { 4 } \ 0 \ \frac { 3 } { 2 } \end{array} \right]$ 
E)  $\left[ \begin{array} { l } x _ { 1 } \ x _ { 2 } \ x _ { 3 } \ x _ { 4 } \end{array} \right] = \left[ \begin{array} { c } 0 \ - \frac { 3 } { 4 } \ 0 \ - \frac { 3 } { 2 } \end{array} \right]$

Question 16

Solve the system of equations algebraically.
$$\left\{ \begin{array} { r } 
x y + 25 = 0 \
49 x - 100 y - 700 = 0
\end{array} \right.$$

Accepted Answer

A)  $\left( 50 , - \frac { 1 } { 2 } \right)$ 
B)  $\left( - 50 , \frac { 1 } { 2 } \right) , \left( 50 , - \frac { 1 } { 2 } \right)$ 
C)  $\left( \frac { 50 } { 7 } , - \frac { 7 } { 2 } \right)$ 
D)  $\left( - \frac { 50 } { 7 } , \frac { 7 } { 2 } \right) , \left( \frac { 50 } { 7 } , - \frac { 7 } { 2 } \right)$ 
E)  no solution

Question 17

Fill in the blank using elementary row operations to form a row-equivalent matrix.
$$\begin{array} { l } 
{ \left[ \begin{array} { c c c } 
2 & - 1 & - 9 \
8 & - 6 & 4
\end{array} \right] } \
{ \left[ \begin{array} { c c c } 
2 & - 1 & - 9 \
0 & \square & 40
\end{array} \right] }
\end{array}$$

Accepted Answer

A)  $\left[ \begin{array} { r r r } 2 & - 1 & - 9 \ 0 & - 2 & 40 \end{array} \right]$ 
B)  $\left[ \begin{array} { c c c } 2 & - 1 & - 9 \ 0 & 0 & 40 \end{array} \right]$ 
C)  $\left[ \begin{array} { c c c } 2 & - 1 & - 9 \ 0 & - 10 & 40 \end{array} \right]$ 
D)  $\left[ \begin{array} { r r r } 2 & - 1 & - 9 \ 0 & - 1 & 40 \end{array} \right]$ 
E)  $\left[ \begin{array} { c c c } 2 & - 1 & - 9 \ 0 & 2 & 40 \end{array} \right]$

Question 18

Use a system of equations to find the specified equation that passes through the points.
Solve the system using matrices.
Parabola: $y = a x ^ { 2 } + b x + c$

Accepted Answer

A) 
$$y = x ^ { 2 } - 3 x + 4$$ 
B) 
$$y = - x ^ { 2 } - 3 x + 1$$ 
C) 
$$y = - x ^ { 2 } + 3 x + 4$$ 
D) 
$$y = - x ^ { 2 } + 3 x + 1$$ 
E) 
$$y = - x ^ { 2 } - 2 x + 1$$

Question 19

Find the minor $M _ { 13 }$ and its cofactor $C _ { 13 }$ of the matrix $\left[ \begin{array} { c c c } 9 & - 6 & 24 \ - 9 & 6 & - 18 \ 3 & - 9 & 18 \end{array} \right]$.

Accepted Answer

A) 
$M _ { 13 } = \left| \begin{array} { l l } 9 & 24 \ 3 & 18 \end{array} \right| = 90$
$C _ { 13 } = - 90$ 
B) 
$M _ { 13 } = \left| \begin{array} { c c } - 6 & 24 \ 6 & - 18 \end{array} \right| = - 36$
$C _ { 13 } = - 36$ 
C) 
$M _ { 13 } = \left| \begin{array} { c c } - 9 & 6 \ 3 & - 9 \end{array} \right| = 63$
$C _ { 13 } = - 63$ 
D) 
$M _ { 13 } = \left| \begin{array} { c c } - 6 & 24 \ 6 & - 18 \end{array} \right| = - 36$
$C _ { 13 } = 36$ 
E) 
$M _ { 13 } = \left| \begin{array} { c c } - 9 & 6 \ 3 & - 9 \end{array} \right| = 63$
$C _ { 13 } = 63$

Question 20

Evaluate the expression.
$\left[ \begin{array} { c } 
- 5 \
- 1 \
5
\end{array} \right] \left( \left[ \begin{array} { l l } 
9 & - 7
\end{array} \right] + \left[ \begin{array} { l l } 
- 7 & - 8
\end{array} \right] \right)$

Accepted Answer

A)  $\left[ \begin{array} { c } - 10 \ - 2 \ 10 \end{array} \right]$ 
B)  $\left[ \begin{array} { c c } - 10 & 75 \ - 2 & 15 \ 10 & - 75 \end{array} \right]$ 
C)  $\left[ \begin{array} { c c c } - 10 & - 2 & 10 \ 75 & 15 & - 75 \end{array} \right]$ 
D)  $\left[ \begin{array} { c c c c } - 45 & 35 & 35 & 40 \ - 9 & 7 & 7 & 8 \ 45 & - 35 & - 35 & - 40 \end{array} \right]$ 
E)  not possible

Exam 7: Linear Systems and Matrices

Determine which one of the ordered triples below is a solution of the given system of equations. $\left\{ \begin{array} { l } 8 x + 6 y - 4 z = 74 \\x - 5 y + 8 z = - 102 \\5 x - 3 y + 2 z = - 46\end{array} \right.$

An object moving vertically is at the given heights at the specified times. Find the position equation $s = \frac { 1 } { 2 } a t ^ { 2 } + v _ { o } t + s _ { o }$ for the object. At $t = 1$ second, $s = 198$ feet At $t = 2$ seconds, $s = 182$ feet At $t = 3$ seconds, $s = 134$ feet

Write the augmented matrix for the system of linear equations. $\left\{ \begin{aligned}x + 4 y - z & = - 9 \\- 9 y + 4 z & = - 6 \\x + 5 z & = - 4\end{aligned} \right.$

Solve using any method. $\left\{ \begin{aligned}7 x + 6 y & = 15 \\y & = x - 7\end{aligned} \right.$

Use a determinant to determine whether the points below are collinear. $( 3 , - 1 ) , ( 0 , - 3 ) , ( 24,13 )$

Solve the system of linear equations. $\left\{ \begin{aligned}2 x - 2 y + 3 z & = - 21 \\2 x + z & = - 11 \\- 2 x + 4 y - 4 z & = 28\end{aligned} \right.$

Write the partial fraction decomposition of the improper rational expression. $\frac { x ^ { 3 } - 4 x ^ { 2 } + 3 x + 2 } { x ^ { 2 } + x - 2 }$

Solve the system by the method of elimination. $\left\{ \begin{array} { c } 5 x - y = 11 \\- 7 x - y = - 25\end{array} \right.$

Find the minor $M _ { 13 }$ and its cofactor $C _ { 13 }$ of the matrix $\left[ \begin{array} { c c c } - 9 & 6 & - 24 \\ 9 & - 6 & 18 \\ - 3 & 9 & - 18 \end{array} \right]$ .

Solve the system of linear equations. $\left\{ \begin{aligned}x + y + z + w & = 11 \\- 2 x - y - 3 z + 4 w & = - 5 \\- 2 x - 5 z - w & = - 23 \\3 x + 5 y + z - 2 w & = 16\end{aligned} \right.$

Solve the system by the method of elimination. Round numbers to three decimal places. $\left\{ \begin{array} { c } 5.7 x + 7.1 y = 3.2 \\- 0.8 x - 4.8 y = - 1.1\end{array} \right.$

Evaluate the determinant $\left| \begin{array} { c c } 7 x & x \ln x \\ 7 & 2 + \ln x \end{array} \right|$ in which the entries are functions.

Find $| A B |$ , if $A = \left[ \begin{array} { r r r } 5 & - 2 & - 2 \\- 3 & 4 & - 1 \\- 5 & 6 & 2\end{array} \right] , B = \left[ \begin{array} { r r r } 1 & - 1 & - 3 \\- 5 & - 3 & 4 \\- 5 & - 1 & - 2\end{array} \right]$

Solve the system of equations algebraically. $\left\{ \begin{array} { r } x y + 25 = 0 \\49 x - 100 y - 700 = 0\end{array} \right.$

Fill in the blank using elementary row operations to form a row-equivalent matrix. $\begin{array} { l } { \left[ \begin{array} { c c c } 2 & - 1 & - 9 \\8 & - 6 & 4\end{array} \right] } \\{ \left[ \begin{array} { c c c } 2 & - 1 & - 9 \\0 & \square & 40\end{array} \right] }\end{array}$

Find the minor $M _ { 13 }$ and its cofactor $C _ { 13 }$ of the matrix $\left[ \begin{array} { c c c } 9 & - 6 & 24 \\ - 9 & 6 & - 18 \\ 3 & - 9 & 18 \end{array} \right]$ .

Evaluate the expression. $\left[ \begin{array} { c } - 5 \\- 1 \\5\end{array} \right] \left( \left[ \begin{array} { l l } 9 & - 7\end{array} \right] + \left[ \begin{array} { l l } - 7 & - 8\end{array} \right] \right)$

Exam 7: Linear Systems and Matrices

Write the augmented matrix for the system of linear equations. {x+4y−z=−9−9y+4z=−6x+5z=−4\left\{ \begin{aligned}x + 4 y - z & = - 9 \\- 9 y + 4 z & = - 6 \\x + 5 z & = - 4\end{aligned} \right.⎩⎨⎧​x+4y−z−9y+4zx+5z​=−9=−6=−4​

Solve using any method. {7x+6y=15y=x−7\left\{ \begin{aligned}7 x + 6 y & = 15 \\y & = x - 7\end{aligned} \right.{7x+6yy​=15=x−7​

Use a determinant to determine whether the points below are collinear. (3,−1) ,(0,−3) ,(24,13) ( 3 , - 1 ) , ( 0 , - 3 ) , ( 24,13 ) (3,−1) ,(0,−3) ,(24,13)

Solve the system of linear equations. {2x−2y+3z=−212x+z=−11−2x+4y−4z=28\left\{ \begin{aligned}2 x - 2 y + 3 z & = - 21 \\2 x + z & = - 11 \\- 2 x + 4 y - 4 z & = 28\end{aligned} \right.⎩⎨⎧​2x−2y+3z2x+z−2x+4y−4z​=−21=−11=28​

Write the partial fraction decomposition of the improper rational expression. x3−4x2+3x+2x2+x−2\frac { x ^ { 3 } - 4 x ^ { 2 } + 3 x + 2 } { x ^ { 2 } + x - 2 }x2+x−2x3−4x2+3x+2​

Solve the system by the method of elimination. {5x−y=11−7x−y=−25\left\{ \begin{array} { c } 5 x - y = 11 \\- 7 x - y = - 25\end{array} \right.{5x−y=11−7x−y=−25​

Find the minor M13M _ { 13 }M13​ and its cofactor C13C _ { 13 }C13​ of the matrix [−96−249−618−39−18]\left[ \begin{array} { c c c } - 9 & 6 & - 24 \\ 9 & - 6 & 18 \\ - 3 & 9 & - 18 \end{array} \right]​−99−3​6−69​−2418−18​​ .

Solve the system by the method of elimination. Round numbers to three decimal places. {5.7x+7.1y=3.2−0.8x−4.8y=−1.1\left\{ \begin{array} { c } 5.7 x + 7.1 y = 3.2 \\- 0.8 x - 4.8 y = - 1.1\end{array} \right.{5.7x+7.1y=3.2−0.8x−4.8y=−1.1​

Evaluate the determinant ∣7xxln⁡x72+ln⁡x∣\left| \begin{array} { c c } 7 x & x \ln x \\ 7 & 2 + \ln x \end{array} \right|​7x7​xlnx2+lnx​​ in which the entries are functions.

Solve the system of equations algebraically. {xy+25=049x−100y−700=0\left\{ \begin{array} { r } x y + 25 = 0 \\49 x - 100 y - 700 = 0\end{array} \right.{xy+25=049x−100y−700=0​

Use a system of equations to find the specified equation that passes through the points. Solve the system using matrices. Parabola: y=ax2+bx+cy = a x ^ { 2 } + b x + cy=ax2+bx+c

Find the minor M13M _ { 13 }M13​ and its cofactor C13C _ { 13 }C13​ of the matrix [9−624−96−183−918]\left[ \begin{array} { c c c } 9 & - 6 & 24 \\ - 9 & 6 & - 18 \\ 3 & - 9 & 18 \end{array} \right]​9−93​−66−9​24−1818​​ .

Evaluate the expression. [−5−15]([9−7]+[−7−8]) \left[ \begin{array} { c } - 5 \\- 1 \\5\end{array} \right] \left( \left[ \begin{array} { l l } 9 & - 7\end{array} \right] + \left[ \begin{array} { l l } - 7 & - 8\end{array} \right] \right) ​−5−15​​([9​−7​]+[−7​−8​])