Question 1

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
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)
$\left[\begin{array}{lll}
2 & 1 & 2 \
0 & 3 & 3 \
7 & 4 & 0
\end{array}\right]$

Accepted Answer

Multiplying row 1 by 2 gives $[4, 2, 4]$. Adding this to row 2 ($[-4, 3, -1]$) results in $[0, 5, 3]$, which matches option C.

Question 2

If the graphs of a system of two equations are both parabolas, what are the possible numbers of solutions (with
real coordinates) of this system?

Accepted Answer

The answer of If the graphs of a system of...

Question 3

Provide an appropriate response.
-When using the substitution or elimination method to solve a system of two equations, you end up with an
equation stating 0 = 0. What does this indicate to you about the system of equations?

Accepted Answer

The answer of Provide an appropriate response.
-When using the substitution...

Question 4

Write the augmented matrix for the system. Do not
-$$\begin{aligned}
6 x + 9 y & = 96 \
- 2 y & = - 16
\end{aligned}$$
A)
$\left[ \begin{array} { r r | r } 96 & 9 & 6 \ - 16 & 0 & - 2 \end{array} \right]$
B)
$\left[ \begin{array} { r r | r } 6 & 9 & 96 \ - 2 & - 16 & 0 \end{array} \right]$
C)
$\left[ \begin{array} { r r | r } 6 & 9 & 96 \ 0 & - 2 & - 16 \end{array} \right]$
D)
$\left[ \begin{array} { r r | r } - 2 & 0 & - 16 \ 6 & 9 & 9 \end{array} \right]$

Accepted Answer

The augmented matrix for a system of equations is formed by writing the coefficients of the variables in each equation as the columns of the matrix, and the constants on the right-hand side of the equations as the last column. The correct matrix for the given system is $\left[ \begin{array} { r r | r } 6 & 9 & 96 \ 0 & - 2 & - 16 \end{array} \right]$, which matches option C.

Question 5

The solution set of the system $y _ { 1 } = - x + 5$ and $y _ { 2 } = - 2 x + 7$ is $\{ ( 2,3 ) \}$. Using slopes and $y$-intercepts, determine which of the two calculator-generated screens, left or right, is the appropriate one for this system.

A) Right
B) Left
C) Neither is correct.
D) Both could be correct.

Accepted Answer

The answer of The solution set of the system \(y...

Question 6

Use the given row transformation to change the matrix as indicated.
-$\left[\begin{array}{rr}
5 & 3 \
10 & 7
\end{array}\right] ;-2 \text { times row } 1 \text { added to row } 2$
A)
$\left[\begin{array}{rr}
5 & 3 \
0 & -1
\end{array}\right]$
B)
$\left[\begin{array}{cr}
-10 & -6 \
0 & 1
\end{array}\right]$
C)
$\left[\begin{array}{ll}
5 & 3 \
1 & 1
\end{array}\right]$
D)
$\left[\begin{array}{ll}
5 & 3 \
0 & 1
\end{array}\right]$

Accepted Answer

-2 times row 1 added to row 2 gives:$\left[\begin{array}{rr}5 & 3 \10 & 7\end{array}\right] \rightarrow \left[\begin{array}{rr}5 & 3 \10-2(5) & 7-2(3)\end{array}\right] = \left[\begin{array}{rr}5 & 3 \0 & 1\end{array}\right]$

Question 7

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
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)
$\left[ \begin{array} { r r r } - 1 & 3 & 8 \ 2 & 1 & 2 \ 4 & - 1 & 4 \end{array} \right]$

Accepted Answer

To apply the transformation, we multiply each element of row 2 by -2 and add the result to the corresponding element of row 3. For row 3, the calculations are as follows: $4 + (-2 \times 2) = 4 - 4 = 0$, $(-1) + (-2 \times 1) = -1 - 2 = -3$, and $4 + (-2 \times 2) = 4 - 4 = 0$. This results in the new row 3 being $0, -3, 0$, which matches option C.

Question 8

Write the augmented matrix for the system. Do not solve the system.
$\begin{array} { l } 
2 x + 3 y = 24 \
4 x + 6 y = 48
\end{array}$
A) $\left[ \begin{array} { l l | l } 
24 & 3 & 2 \
48 & 4 & 6
\end{array} \right]$
B) $\left[ \begin{array} { l l | l } 
2 & 4 & 24 \
3 & 6 & 48
\end{array} \right]$
C) $\left[ \begin{array} { l l | l } 
2 & 3 & 48 \
6 & 4 & 24
\end{array} \right]$
D) $\left[ \begin{array} { l l | l } 
2 & 3 & 24 \
4 & 6 & 48
\end{array} \right]$

Accepted Answer

The augmented matrix correctly represents the system of equations with coefficients of $x$ and $y$ in the first two columns and the constants on the right side of the equation in the third column.

Question 9

Write the augmented matrix for the system. Do not
-$$\begin{array} { l } 
7 x + 4 y = 39 \
5 x + 3 y = 29
\end{array}$$
A)
$$\left[ \begin{array} { l l | l } 
7 & 5 & 39 \
4 & 3 & 29
\end{array} \right]$$
B)
$\left[\begin{array}{ll|l}
7 & 4 & 39 \
5 & 3 & 29
\end{array}\right]$
C)
$$\left[ \begin{array} { l l | l } 
39 & 4 & 7 \
29 & 5 & 3
\end{array} \right]$$
D)
$$\left[ \begin{array} { l l | l } 
7 & 4 & 29 \
3 & 5 & 39
\end{array} \right]$$

Accepted Answer

The augmented matrix correctly represents the system of equations, with the coefficients of $x$ and $y$ in the first two columns and the constants on the right side of the bar.

Question 10

If the graphs of a system of two equations are a line and a parabola, what are the possible numbers of solutions
(with real coordinates) of this system?

Accepted Answer

The answer of If the graphs of a system of...

Question 11

Provide an appropriate response.
-$$\begin{array} { l } 
2 x - 7 y = - 17 \
5 x + 3 y = 19
\end{array}$$ If your friend were going to solve this system of equations by first eliminating y, what general suggestions
would you make so your friend could start on this in a systematic way?

Accepted Answer

The answer of Provide an appropriate response.
-\[\begin{array} { l }...

Question 12

Use the given row transformation to change the matrix as indicated.
-$\left[ \begin{array} { r r } - 1 & 4 \ 7 & 0 \end{array} \right] ; 7$ times row 1 added to row 2
A)
$\left[ \begin{array} { r r } - 1 & 4 \ 0 & 28 \end{array} \right]$
B)

$\left[ \begin{array} { r r } - 1 & - 3 \ 0 & 28 \end{array} \right]$
C)
$\left[ \begin{array} { r r } - 7 & 28 \ 7 & 0 \end{array} \right]$
D)
$\left[ \begin{array} { r r } - 1 & 4 \ 7 & 28 \end{array} \right]$

Accepted Answer

To perform the row operation, multiply each element of row 1 by 7 and add it to the corresponding element of row 2. For the first element: $7 \times (-1) + 7 = 0$. For the second element: $7 \times 4 + 0 = 28$. This results in the new row 2 being [0, 28], making option A correct.

Question 13

If the graphs of a system of two equations are a circle and a parabola, what are the possible numbers of
solutions (with real coordinates) of this system?

Accepted Answer

The answer of If the graphs of a system of...

Question 14

Provide an appropriate response.
-When using the substitution or elimination method to solve a system of two equations, you end up with an
equation stating $0 = 7$ What does this indicate to you about the system of equations?

Accepted Answer

The answer of Provide an appropriate response.
-When using the substitution...

Question 15

If the graphs of a system of two equations are both circles, what are the possible numbers of solutions (with real
coordinates) of this system?

Accepted Answer

The answer of If the graphs of a system of...

Question 16

Write the augmented matrix for the system. Do not
-$$\begin{array} { l } 
6 x + 7 y + 8 z = 63 \
5 x + 3 y + 2 z = 46 \
4 x + 9 y - 2 z = 23
\end{array}$$
A)
$$\left[ \begin{array} { r r r | r } 
6 & 5 & 4 & 63 \
7 & 3 & 9 & 46 \
8 & 2 & - 2 & 23
\end{array} \right]$$
B)
$\left[ \begin{array} { r r r | r } 6 & 7 & 8 & 63 \ 5 & 3 & 2 & 46 \ 4 & 9 & - 2 & 23 \end{array} \right]$
C)
$\left[ \begin{array} { r r r | r } 63 & 8 & 7 & 6 \ 46 & 2 & 3 & 5 \ 23 & - 2 & 9 & 4 \end{array} \right]$
D)
$\left[ \begin{array} { r r r } 6 & 7 & 8 \ 5 & 3 & 2 \ 4 & 9 & - 2 \end{array} \right]$

Accepted Answer

The correct augmented matrix for the given system of equations aligns the coefficients of $x$, $y$, and $z$ in the first three columns, respectively, and the constants on the right side of the equation in the fourth column, exactly as shown in option B.

Question 17

The solution set of the system $y _ { 1 } = 2 x + 3$ and $y _ { 2 } = - x + 9$ is $\{ ( 2,7 ) \}$. Using slopes and $y$-intercepts, determine which of the two calculator-generated screens, left or right, is the appropriate one for this system.

A) Right
B) Left
C) Neither is correct.
D) Both could be correct.

Accepted Answer

The answer of The solution set of the system \(y...

Question 18

Provide an appropriate response.
-$$\begin{array} { l } 
5 x + 3 y = 19 \
7 x + 8 y = 48
\end{array}$$ If your friend were going to solve this system of equations by first eliminating y, what general suggestions
would you make so your friend could start on this in a systematic way?

Accepted Answer

The answer of Provide an appropriate response.
-\[\begin{array} { l }...

Question 19

Use the given row transformation to change the matrix as indicated.
-$\left[ \begin{array} { r r r } 6 & 5 & 18 \ 5 & - 2 & 7 \ 1 & 2 & 3 \end{array} \right] ; - 6$ times row 3 added to row 1
A)
$\left[ \begin{array} { r r r } 6 & 5 & 18 \ - 1 & - 14 & - 11 \ - 6 & - 12 & - 18 \end{array} \right]$
B)
$\left[ \begin{array} { r r r } 7 & 7 & 0 \ 5 & - 2 & 7 \ 1 & 2 & 3 \end{array} \right]$
C)
$\left[\begin{array}{rrr}
0 & -7 & 0 \
5 & -2 & 7 \
1 & 2 & 3
\end{array}\right]$
D)
$\left[\begin{array}{rrr}
6 & 5 & 1 \
5 & -2 & 7 \
-6 & -12 & -18
\end{array}\right]$

Accepted Answer

The answer of Use the given row transformation to change...

Question 20

Write the augmented matrix for the system. Do not
-$$\begin{array} { r } 
6 x + 2 z = 10 \
4 y + 7 z = 26 \
9 x + 3 y + 7 z = 32
\end{array}$$
A)
$$\left[ \begin{array} { l l l | l } 
6 & 2 & 0 & 10 \
4 & 7 & 0 & 26 \
9 & 3 & 7 & 32
\end{array} \right]$$
B)
$\left[ \begin{array} { l l l } 6 & 0 & 2 \ 0 & 4 & 7 \ 9 & 3 & 7 \end{array} \right]$
C)
$\left[ \begin{array} { l l l | l } 6 & 0 & 9 & 10 \ 0 & 4 & 3 & 26 \ 2 & 7 & 7 & 32 \end{array} \right]$
D)
$\left[ \begin{array} { l l l | l } 6 & 0 & 2 & 10 \ 0 & 4 & 7 & 26 \ 9 & 3 & 7 & 32 \end{array} \right]$

Accepted Answer

The augmented matrix correctly represents the coefficients of the variables and the constants from the system of equations.

Quiz 6: Systems and Matrices

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

If the graphs of a system of two equations are both parabolas, what are the possible numbers of solutions (with real coordinates) of this system?

Provide an appropriate response. -When using the substitution or elimination method to solve a system of two equations, you end up with an equation stating 0 = 0. What does this indicate to you about the system of equations?

Write the augmented matrix for the system. Do not - $\begin{aligned}6 x + 9 y & = 96 \\- 2 y & = - 16\end{aligned}$

Use the given row transformation to change the matrix as indicated. - $\left[\begin{array}{rr}5 & 3 \\10 & 7\end{array}\right] ;-2 \text { times row } 1 \text { added to row } 2$

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

Write the augmented matrix for the system. Do not solve the system. $\begin{array} { l } 2 x + 3 y = 24 \\4 x + 6 y = 48\end{array}$

Write the augmented matrix for the system. Do not - $\begin{array} { l } 7 x + 4 y = 39 \\5 x + 3 y = 29\end{array}$

If the graphs of a system of two equations are a line and a parabola, what are the possible numbers of solutions (with real coordinates) of this system?

Provide an appropriate response. - $\begin{array} { l } 2 x - 7 y = - 17 \\5 x + 3 y = 19\end{array}$ If your friend were going to solve this system of equations by first eliminating y, what general suggestions would you make so your friend could start on this in a systematic way?

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

If the graphs of a system of two equations are a circle and a parabola, what are the possible numbers of solutions (with real coordinates) of this system?

Provide an appropriate response. -When using the substitution or elimination method to solve a system of two equations, you end up with an equation stating $0 = 7$ What does this indicate to you about the system of equations?

If the graphs of a system of two equations are both circles, what are the possible numbers of solutions (with real coordinates) of this system?

Write the augmented matrix for the system. Do not - $\begin{array} { l } 6 x + 7 y + 8 z = 63 \\5 x + 3 y + 2 z = 46 \\4 x + 9 y - 2 z = 23\end{array}$

Provide an appropriate response. - $\begin{array} { l } 5 x + 3 y = 19 \\7 x + 8 y = 48\end{array}$ If your friend were going to solve this system of equations by first eliminating y, what general suggestions would you make so your friend could start on this in a systematic way?

Use the given row transformation to change the matrix as indicated. - $\left[ \begin{array} { r r r } 6 & 5 & 18 \\ 5 & - 2 & 7 \\ 1 & 2 & 3 \end{array} \right] ; - 6$ times row 3 added to row 1

Write the augmented matrix for the system. Do not - $\begin{array} { r } 6 x + 2 z = 10 \\4 y + 7 z = 26 \\9 x + 3 y + 7 z = 32\end{array}$

Quiz 6: Systems and Matrices

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

If the graphs of a system of two equations are both parabolas, what are the possible numbers of solutions (with real coordinates) of this system?

Provide an appropriate response. -When using the substitution or elimination method to solve a system of two equations, you end up with an equation stating 0 = 0. What does this indicate to you about the system of equations?

Write the augmented matrix for the system. Do not - 6x+9y=96−2y=−16\begin{aligned}6 x + 9 y & = 96 \\- 2 y & = - 16\end{aligned}6x+9y−2y​=96=−16​

Use the given row transformation to change the matrix as indicated. - [53107];−2 times row 1 added to row 2\left[\begin{array}{rr}5 & 3 \\10 & 7\end{array}\right] ;-2 \text { times row } 1 \text { added to row } 2[510​37​];−2 times row 1 added to row 2

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

Write the augmented matrix for the system. Do not solve the system. 2x+3y=244x+6y=48\begin{array} { l } 2 x + 3 y = 24 \\4 x + 6 y = 48\end{array}2x+3y=244x+6y=48​

Write the augmented matrix for the system. Do not - 7x+4y=395x+3y=29\begin{array} { l } 7 x + 4 y = 39 \\5 x + 3 y = 29\end{array}7x+4y=395x+3y=29​

If the graphs of a system of two equations are a line and a parabola, what are the possible numbers of solutions (with real coordinates) of this system?

Use the given row transformation to change the matrix as indicated. - [−1470];7\left[ \begin{array} { r r } - 1 & 4 \\ 7 & 0 \end{array} \right] ; 7[−17​40​];7 times row 1 added to row 2

If the graphs of a system of two equations are a circle and a parabola, what are the possible numbers of solutions (with real coordinates) of this system?

Provide an appropriate response. -When using the substitution or elimination method to solve a system of two equations, you end up with an equation stating 0=70 = 70=7 What does this indicate to you about the system of equations?

If the graphs of a system of two equations are both circles, what are the possible numbers of solutions (with real coordinates) of this system?

Write the augmented matrix for the system. Do not - 6x+7y+8z=635x+3y+2z=464x+9y−2z=23\begin{array} { l } 6 x + 7 y + 8 z = 63 \\5 x + 3 y + 2 z = 46 \\4 x + 9 y - 2 z = 23\end{array}6x+7y+8z=635x+3y+2z=464x+9y−2z=23​

The solution set of the system y1=2x+3y _ { 1 } = 2 x + 3y1​=2x+3 and y2=−x+9y _ { 2 } = - x + 9y2​=−x+9 is {(2,7) }\{ ( 2,7 ) \}{(2,7) } . Using slopes and yyy -intercepts, determine which of the two calculator-generated screens, left or right, is the appropriate one for this system.

Use the given row transformation to change the matrix as indicated. - [65185−27123];−6\left[ \begin{array} { r r r } 6 & 5 & 18 \\ 5 & - 2 & 7 \\ 1 & 2 & 3 \end{array} \right] ; - 6​651​5−22​1873​​;−6 times row 3 added to row 1

Write the augmented matrix for the system. Do not - 6x+2z=104y+7z=269x+3y+7z=32\begin{array} { r } 6 x + 2 z = 10 \\4 y + 7 z = 26 \\9 x + 3 y + 7 z = 32\end{array}6x+2z=104y+7z=269x+3y+7z=32​

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

Write the augmented matrix for the system. Do not - $\begin{aligned}6 x + 9 y & = 96 \\- 2 y & = - 16\end{aligned}$

Use the given row transformation to change the matrix as indicated. - $\left[\begin{array}{rr}5 & 3 \\10 & 7\end{array}\right] ;-2 \text { times row } 1 \text { added to row } 2$

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

Write the augmented matrix for the system. Do not solve the system. $\begin{array} { l } 2 x + 3 y = 24 \\4 x + 6 y = 48\end{array}$

Write the augmented matrix for the system. Do not - $\begin{array} { l } 7 x + 4 y = 39 \\5 x + 3 y = 29\end{array}$

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

Provide an appropriate response. -When using the substitution or elimination method to solve a system of two equations, you end up with an equation stating $0 = 7$ What does this indicate to you about the system of equations?

Write the augmented matrix for the system. Do not - $\begin{array} { l } 6 x + 7 y + 8 z = 63 \\5 x + 3 y + 2 z = 46 \\4 x + 9 y - 2 z = 23\end{array}$

Use the given row transformation to change the matrix as indicated. - $\left[ \begin{array} { r r r } 6 & 5 & 18 \\ 5 & - 2 & 7 \\ 1 & 2 & 3 \end{array} \right] ; - 6$ times row 3 added to row 1

Write the augmented matrix for the system. Do not - $\begin{array} { r } 6 x + 2 z = 10 \\4 y + 7 z = 26 \\9 x + 3 y + 7 z = 32\end{array}$