Question 1

Use the Gauss-Jordan method to solve the system of equations.
-$9 x+6 y-z=60$
$x+5 y+3 z=55$
$-7 \mathrm{x}+\mathrm{y}+\mathrm{z}=5$

Accepted Answer

A)  No solution 
B)  $(1,9,3)$ 
C)  $(1,3,9)$ 
D)  $(-1,9,2)$ 
A)  No solution 
B)  $(1,9,3)$ 
C)  $(1,3,9)$ 
D)  $(-1,9,2)$

Question 2

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,dependent, or inconsistent.
-$x+5 y+5 z=-9$
$2 y+2 z=-2$
$\mathrm{z}=4$

Accepted Answer

A)  Independent 
B)  Inconsistent 
C)  Dependent 
A)  Independent 
B)  Inconsistent 
C)  Dependent

Question 3

Solve the problem.
-Three different high schools plan to order the same three text books. School A plans to order 20 of book 1, 100 of book 2, 80 of book 3. School B plans to order 40 of book 1, 40 of book 2, 80 of book 3. School C plans to order 70 of book 1, 40 of book 2, 10 of book 3. The cost of book 1 is $\$ 15$ per copy, the cost of book 2 is $\$ 20$ per copy, and the cost of book 3 is $\$ 25$ per copy. What matrix product displays the cost to each school of buying the textbooks? Display the two matrices which must be multiplied and their product.

Accepted Answer

A)   
B)  $\left[\begin{array}{lll}15 & 20 & 25\end{array}\right]\left[\begin{array}{l}200 \ 160 \ 120\end{array}\right]=\left[\begin{array}{lll}300032003000\end{array}\right]$ 
C) $\left[\begin{array}{lll}15 & 20 & 25\end{array}\right]\left[\begin{array}{lll}20 & 100 & 80 \ 40 & 40 & 80 \ 70 & 40 & 10\end{array}\right]=\left[\begin{array}{lll}2850 & 3300 & 3050\end{array}\right]$ 
D) $\left[\begin{array}{lll}15 & 20 & 25\end{array}\right]\left[\begin{array}{l}130 \ 180 \ 170\end{array}\right]=\left[\begin{array}{ll}195036004250\end{array}\right]$ 
A)   
B)  $\left[\begin{array}{lll}15 & 20 & 25\end{array}\right]\left[\begin{array}{l}200 \ 160 \ 120\end{array}\right]=\left[\begin{array}{lll}300032003000\end{array}\right]$ 
C) $\left[\begin{array}{lll}15 & 20 & 25\end{array}\right]\left[\begin{array}{lll}20 & 100 & 80 \ 40 & 40 & 80 \ 70 & 40 & 10\end{array}\right]=\left[\begin{array}{lll}2850 & 3300 & 3050\end{array}\right]$ 
D) $\left[\begin{array}{lll}15 & 20 & 25\end{array}\right]\left[\begin{array}{l}130 \ 180 \ 170\end{array}\right]=\left[\begin{array}{ll}195036004250\end{array}\right]$

Question 4

Multiply both sides of each equation by a common denominator to eliminate the fractions. Then solve the system.
-$\frac{1}{5} x+\frac{1}{5} y=-1$
$x-y=-9$

Accepted Answer

A)  $(-8,3)$ 
B)  No solution 
C)  $(-7,2)$ 
D)  $(7,3)$ 
A)  $(-8,3)$ 
B)  No solution 
C)  $(-7,2)$ 
D)  $(7,3)$

Question 5

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,dependent, or inconsistent.
-$x-y+z=0$
$x+y+z=-10$
$x+y-z=-2$

Accepted Answer

A)  Dependent 
B)  Independent 
C)  Inconsistent 
A)  Dependent 
B)  Independent 
C)  Inconsistent

Question 6

Solve the matrix equation for $X$.}
-$\mathrm{M}=\left[\begin{array}{rr}-5 & 2 \ 1 & 4\end{array}\right], \mathrm{N}=\left[\begin{array}{rr}-14 & 14 \ -2 & -20\end{array}\right], \mathrm{N}=\mathrm{MX}+\mathrm{X}$

Accepted Answer

A) $\left[\begin{array}{rr}2 & -5 \ -1 & -2\end{array}\right]$ 
B) $\left[\begin{array}{rr}4 & -6 \ -2 & -3\end{array}\right]$ 
C) $\left[\begin{array}{rr}3 & -5 \ -1 & -3\end{array}\right]$ 
D)  $\left[\begin{array}{ll}3 & -4 \ 0 & -4\end{array}\right]$ 
A) $\left[\begin{array}{rr}2 & -5 \ -1 & -2\end{array}\right]$ 
B) $\left[\begin{array}{rr}4 & -6 \ -2 & -3\end{array}\right]$ 
C) $\left[\begin{array}{rr}3 & -5 \ -1 & -3\end{array}\right]$ 
D)  $\left[\begin{array}{ll}3 & -4 \ 0 & -4\end{array}\right]$

Question 7

Perform the indicated operation where possible.
-$\left[\begin{array}{lll}-1 & 9 & 3\end{array}\right]-\left[\begin{array}{ll}4 & 3\end{array}\right]$

Accepted Answer

A)  $\left[\begin{array}{lll}-5 & 6 & 0\end{array}\right]$ 
B)  Not defined 
C)  $\left[\begin{array}{lll}-5 & 9 & 0\end{array}\right]$ 
D)  $\left[\begin{array}{lll}-5 & 6 & 3\end{array}\right]$ 
A)  $\left[\begin{array}{lll}-5 & 6 & 0\end{array}\right]$ 
B)  Not defined 
C)  $\left[\begin{array}{lll}-5 & 9 & 0\end{array}\right]$ 
D)  $\left[\begin{array}{lll}-5 & 6 & 3\end{array}\right]$

Question 8

Solve the problem by writing and solving a suitable system of equations.
-A chemistry department wants to make 3 liters of a $17.5 \%$ basic solution by mixing a $20 \%$ solution with a $15 \%$ solution. How many liters of each type of basic solution should be used to produce the $17.5 \%$ solution?

Accepted Answer

A)  1 liter of $15 \%$ and 2 liters of $20 \%$ 
B)  1.5 liters of each 
C)  2 liters of $15 \%$ and 1 liter of $20 \%$ 
D)  0.5 liter of $15 \%$ and 2.5 liters of $20 \%$ 
A)  1 liter of $15 \%$ and 2 liters of $20 \%$ 
B)  1.5 liters of each 
C)  2 liters of $15 \%$ and 1 liter of $20 \%$ 
D)  0.5 liter of $15 \%$ and 2.5 liters of $20 \%$

Question 9

The reduced row echelon form of the augmented matrix of a system of equations is given. Find the solutions of thesystem.
-45 $\left[\begin{array}{llll|c}1 & 0 & 0 & 0 & 12 \ 0 & 1 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 & -8 \ 0 & 0 & 0 & 1 & 11 / 2\end{array}\right]$

Accepted Answer

A)  $\left(12, w,-8, \frac{11}{2}\right)$ for any real number $w$ 
B)  $\left(12,0,-8, \frac{11}{2}\right)$ 
C)  $\left(12,-8, \frac{11}{2}, 0\right)$ 
D)  No solution 
A)  $\left(12, w,-8, \frac{11}{2}\right)$ for any real number $w$ 
B)  $\left(12,0,-8, \frac{11}{2}\right)$ 
C)  $\left(12,-8, \frac{11}{2}, 0\right)$ 
D)  No solution

Question 10

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

Accepted Answer

A) $\left[\begin{array}{r}4 \ -6 \ 4\end{array}\right]$ 
B) $\left[\begin{array}{r}-4 \ 12 \ -8\end{array}\right]$ 
C) $\left[\begin{array}{r}-2 \ 6 \ -4\end{array}\right]$ 
D) $\left[\begin{array}{r}4 \ -12 \ 8\end{array}\right]$ 
A) $\left[\begin{array}{r}4 \ -6 \ 4\end{array}\right]$ 
B) $\left[\begin{array}{r}-4 \ 12 \ -8\end{array}\right]$ 
C) $\left[\begin{array}{r}-2 \ 6 \ -4\end{array}\right]$ 
D) $\left[\begin{array}{r}4 \ -12 \ 8\end{array}\right]$

Question 11

Multiply both sides of each equation by a common denominator to eliminate the fractions. Then solve the system.
-$\frac{1}{3} x+\frac{1}{3} y=0$
$\frac{1}{3} \mathrm{x}-\frac{1}{3} \mathrm{y}=\frac{4}{3}$

Accepted Answer

A)  $(1,-1)$ 
B)  $(2,-2)$ 
C)  $(-2,-1)$ 
D)  No solution 
A)  $(1,-1)$ 
B)  $(2,-2)$ 
C)  $(-2,-1)$ 
D)  No solution

Question 12

Write the word or phrase that best completes each statement or answers the question
-Using the matrix $L=\left[\begin{array}{ccc}1 & m & n \ p & q & r\end{array}\right]$, verify that $L+(-L)=O$.

Accepted Answer

The answer of Write the word or phrase that best...

Question 13

Solve the problem by writing and solving a suitable system of equations.
-A store sells televisions for $\$ 360$ and video cassette recorders for $\$ 270$. At the beginning of the week its entire stock is worth $\$ 51,570$. During the week it sells three quarters of the televisions and one third of the video cassette recorders for a total of $\$ 31,590$. How many televisions and video cassette recorders did it have in its stock at the beginning of the week?

Accepted Answer

A)  94 televisions; 69 video cassette recorders 
B)  95 televisions; 66 video cassette recorders 
C)  96 televisions; 63 video cassette recorders 
D)  93 televisions; 67 video cassette recorders 
A)  94 televisions; 69 video cassette recorders 
B)  95 televisions; 66 video cassette recorders 
C)  96 televisions; 63 video cassette recorders 
D)  93 televisions; 67 video cassette recorders

Question 14

The reduced row echelon form of the augmented matrix of a system of equations is given. Find the solutions of thesystem.
-$\left[\begin{array}{rrrr|r}1 & 0 & 0 & 0 & -2 \ 0 & 1 & 0 & 0 & 19 \ 0 & 0 & 1 & 0 & 6 \ 0 & 0 & 0 & 1 & 2 \ 0 & 0 & 0 & 0 & 0\end{array}\right]$

Accepted Answer

A)  $(-2,19,6,2, w)$ for any real number $\mathrm{w}$ 
B)  $(-2,19,6,2)$ 
C)  $(-2,19,6,2,0)$ 
D)  No solution 
A)  $(-2,19,6,2, w)$ for any real number $\mathrm{w}$ 
B)  $(-2,19,6,2)$ 
C)  $(-2,19,6,2,0)$ 
D)  No solution

Question 15

Write a matrix to display the information.
-At a store, Sam bought 4 batteries, 1460 -watt light bulbs, 44100 -watt light bulbs, 5 picture-hanging kits, and a hammer. Jennifer bought 9 batteries, 2100 -watt light bulbs, and a package of tacks. Write the information as a $2 \times 6$ matrix.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 16

Solve the matrix equation for $X$.}
-$\mathrm{M}=\left[\begin{array}{rr}6 & -5 \ 2 & 6\end{array}\right], \mathrm{N}=\left[\begin{array}{r}-10 \ 24\end{array}\right], \mathrm{N}=\mathrm{X}-\mathrm{MX}$

Accepted Answer

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

Question 17

Use the Gauss-Jordan method to solve the system of equations.
-$x-y+z=-7$
$x+y+z=1$
$\mathrm{x}+\mathrm{y}-\mathrm{z}=11$

Accepted Answer

A)  $(2,-5,4)$ 
B)  $(-5,2,4)$ 
C)  No solution 
D)  $(2,4,-5)$ 
A)  $(2,-5,4)$ 
B)  $(-5,2,4)$ 
C)  No solution 
D)  $(2,4,-5)$

Question 18

Determine whether the given ordered set of numbers is a solution of the system of equations.
-$(-2,-1,3)$
$4 x-4 y+z=-1$
$5 \mathrm{x}+5 \mathrm{z}=5$
$x+4 y-2 z=-12$

Accepted Answer

A) True 
 B)False

Question 19

Determine whether the two matrices are inverses of each other by computing their product.
-$\left[\begin{array}{ll}-5 & 1 \ -7 & 1\end{array}\right],\left[\begin{array}{ll}\frac{1}{2} & -\frac{1}{2} \ \frac{7}{2} & -\frac{5}{2}\end{array}\right]$

Accepted Answer

A) True 
 B)False

Question 20

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter $\mathrm{z}$.
-$x-y+4 z=-22$
$4 \mathrm{x}+\mathrm{z}=-5$
$x+3 y+z=1$

Accepted Answer

A)  $(-5,0,2)$ 
B)  No solution 
C)  $(-5,2,0)$ 
D)  $(0,2,-5)$ 
A)  $(-5,0,2)$ 
B)  No solution 
C)  $(-5,2,0)$ 
D)  $(0,2,-5)$

Use the Gauss-Jordan method to solve the system of equations. - $9 x+6 y-z=60$ $x+5 y+3 z=55$ $-7 \mathrm{x}+\mathrm{y}+\mathrm{z}=5$

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,dependent, or inconsistent. - $x+5 y+5 z=-9$ $2 y+2 z=-2$ $\mathrm{z}=4$

Multiply both sides of each equation by a common denominator to eliminate the fractions. Then solve the system. - $\frac{1}{5} x+\frac{1}{5} y=-1$ $x-y=-9$

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,dependent, or inconsistent. - $x-y+z=0$ $x+y+z=-10$ $x+y-z=-2$

Solve the matrix equation for $X$ .} - $\mathrm{M}=\left[\begin{array}{rr}-5 & 2 \\ 1 & 4\end{array}\right], \mathrm{N}=\left[\begin{array}{rr}-14 & 14 \\ -2 & -20\end{array}\right], \mathrm{N}=\mathrm{MX}+\mathrm{X}$

Perform the indicated operation where possible. - $\left[\begin{array}{lll}-1 & 9 & 3\end{array}\right]-\left[\begin{array}{ll}4 & 3\end{array}\right]$

Solve the problem by writing and solving a suitable system of equations. -A chemistry department wants to make 3 liters of a $17.5 \%$ basic solution by mixing a $20 \%$ solution with a $15 \%$ solution. How many liters of each type of basic solution should be used to produce the $17.5 \%$ solution?

The reduced row echelon form of the augmented matrix of a system of equations is given. Find the solutions of thesystem. -45 $\left[\begin{array}{llll|c}1 & 0 & 0 & 0 & 12 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -8 \\ 0 & 0 & 0 & 1 & 11 / 2\end{array}\right]$

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

Multiply both sides of each equation by a common denominator to eliminate the fractions. Then solve the system. - $\frac{1}{3} x+\frac{1}{3} y=0$ $\frac{1}{3} \mathrm{x}-\frac{1}{3} \mathrm{y}=\frac{4}{3}$

Write the word or phrase that best completes each statement or answers the question -Using the matrix $L=\left[\begin{array}{ccc}1 & m & n \\ p & q & r\end{array}\right]$ , verify that $L+(-L)=O$ .

The reduced row echelon form of the augmented matrix of a system of equations is given. Find the solutions of thesystem. - $\left[\begin{array}{rrrr|r}1 & 0 & 0 & 0 & -2 \\ 0 & 1 & 0 & 0 & 19 \\ 0 & 0 & 1 & 0 & 6 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]$

Write a matrix to display the information. -At a store, Sam bought 4 batteries, 1460 -watt light bulbs, 44100 -watt light bulbs, 5 picture-hanging kits, and a hammer. Jennifer bought 9 batteries, 2100 -watt light bulbs, and a package of tacks. Write the information as a $2 \times 6$ matrix.

Solve the matrix equation for $X$ .} - $\mathrm{M}=\left[\begin{array}{rr}6 & -5 \\ 2 & 6\end{array}\right], \mathrm{N}=\left[\begin{array}{r}-10 \\ 24\end{array}\right], \mathrm{N}=\mathrm{X}-\mathrm{MX}$

Use the Gauss-Jordan method to solve the system of equations. - $x-y+z=-7$ $x+y+z=1$ $\mathrm{x}+\mathrm{y}-\mathrm{z}=11$

Determine whether the given ordered set of numbers is a solution of the system of equations. - $(-2,-1,3)$ $4 x-4 y+z=-1$ $5 \mathrm{x}+5 \mathrm{z}=5$ $x+4 y-2 z=-12$

Determine whether the two matrices are inverses of each other by computing their product. - $\left[\begin{array}{ll}-5 & 1 \\ -7 & 1\end{array}\right],\left[\begin{array}{ll}\frac{1}{2} & -\frac{1}{2} \\ \frac{7}{2} & -\frac{5}{2}\end{array}\right]$

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter $\mathrm{z}$ . - $x-y+4 z=-22$ $4 \mathrm{x}+\mathrm{z}=-5$ $x+3 y+z=1$

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Exam 6: Systems of Linear Equations and Matrices

Use the Gauss-Jordan method to solve the system of equations. - 9x+6y−z=609 x+6 y-z=609x+6y−z=60 x+5y+3z=55x+5 y+3 z=55x+5y+3z=55 −7x+y+z=5-7 \mathrm{x}+\mathrm{y}+\mathrm{z}=5−7x+y+z=5

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,dependent, or inconsistent. - x+5y+5z=−9x+5 y+5 z=-9x+5y+5z=−9 2y+2z=−22 y+2 z=-22y+2z=−2 z=4\mathrm{z}=4z=4

Multiply both sides of each equation by a common denominator to eliminate the fractions. Then solve the system. - 15x+15y=−1\frac{1}{5} x+\frac{1}{5} y=-151​x+51​y=−1 x−y=−9x-y=-9x−y=−9

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,dependent, or inconsistent. - x−y+z=0x-y+z=0x−y+z=0 x+y+z=−10x+y+z=-10x+y+z=−10 x+y−z=−2x+y-z=-2x+y−z=−2

Perform the indicated operation where possible. - [−193]−[43]\left[\begin{array}{lll}-1 & 9 & 3\end{array}\right]-\left[\begin{array}{ll}4 & 3\end{array}\right][−1​9​3​]−[4​3​]

Perform the indicated operation. - Let⁡C=[1−32]\operatorname{Let} C=\left[\begin{array}{r}1 \\ -3 \\ 2\end{array}\right]LetC=​1−32​​ and D=[−13−2]D=\left[\begin{array}{r}-1 \\ 3 \\ -2\end{array}\right]D=​−13−2​​ . Find CCC - 3D.

Multiply both sides of each equation by a common denominator to eliminate the fractions. Then solve the system. - 13x+13y=0\frac{1}{3} x+\frac{1}{3} y=031​x+31​y=0 13x−13y=43\frac{1}{3} \mathrm{x}-\frac{1}{3} \mathrm{y}=\frac{4}{3}31​x−31​y=34​

Write the word or phrase that best completes each statement or answers the question -Using the matrix L=[1mnpqr]L=\left[\begin{array}{ccc}1 & m & n \\ p & q & r\end{array}\right]L=[1p​mq​nr​] , verify that L+(−L)=OL+(-L)=OL+(−L)=O .

Write a matrix to display the information. -At a store, Sam bought 4 batteries, 1460 -watt light bulbs, 44100 -watt light bulbs, 5 picture-hanging kits, and a hammer. Jennifer bought 9 batteries, 2100 -watt light bulbs, and a package of tacks. Write the information as a 2×62 \times 62×6 matrix.

Solve the matrix equation for XXX .} - M=[6−526],N=[−1024],N=X−MX\mathrm{M}=\left[\begin{array}{rr}6 & -5 \\ 2 & 6\end{array}\right], \mathrm{N}=\left[\begin{array}{r}-10 \\ 24\end{array}\right], \mathrm{N}=\mathrm{X}-\mathrm{MX}M=[62​−56​],N=[−1024​],N=X−MX

Use the Gauss-Jordan method to solve the system of equations. - x−y+z=−7x-y+z=-7x−y+z=−7 x+y+z=1x+y+z=1x+y+z=1 x+y−z=11\mathrm{x}+\mathrm{y}-\mathrm{z}=11x+y−z=11

Determine whether the given ordered set of numbers is a solution of the system of equations. - (−2,−1,3)(-2,-1,3)(−2,−1,3) 4x−4y+z=−14 x-4 y+z=-14x−4y+z=−1 5x+5z=55 \mathrm{x}+5 \mathrm{z}=55x+5z=5 x+4y−2z=−12x+4 y-2 z=-12x+4y−2z=−12

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter z\mathrm{z}z . - x−y+4z=−22x-y+4 z=-22x−y+4z=−22 4x+z=−54 \mathrm{x}+\mathrm{z}=-54x+z=−5 x+3y+z=1x+3 y+z=1x+3y+z=1