Write the system of equations in a matrix:

Question

2x + y = 30

x + 4z = 20

Y + 2z = 10

(1 point)
Responses

⎡⎣⎢200042302010⎤⎦⎥
[
2
0
30
0
4
20
0
2
10
]
the 3 by 3 matrix row 1, 2 0 30 row 2, 0 4 20 row 3, 0 2 10

⎡⎣⎢210101042302010⎤⎦⎥
[
2
1
0
30
1
0
4
20
0
1
2
10
]
the 3 by 4 matrix, row 1, column 1, 2 column 2, 1 column 3, 0 column 4, 30 row 2, column 1, 1 column 2, 0 column 3, 4 column 4, 20 row 3, column 1, 0 column 2, 1 column 3, 2 column 4, 10

⎡⎣⎢211142302010⎤⎦⎥
[
2
1
30
1
4
20
1
2
10
]
the 3 by 3 matrix row 1, 2 1 30 row 2, 1 4 20 row 3, 1 2 10

⎡⎣⎢200000042302010⎤⎦⎥

Answer 1

To write the given system of equations in matrix form, we separate the coefficients of the variables \(x\), \(y\), and \(z\) along with the constants on the right-hand side.

The equations given are:

1. \(2x + 1y + 0z = 30\)
2. \(1x + 0y + 4z = 20\)
3. \(0x + 1y + 2z = 10\)

The corresponding matrix form can be represented as:

\[ \begin{bmatrix} 2 & 1 & 0 & | & 30 \ 1 & 0 & 4 & | & 20 \ 0 & 1 & 2 & | & 10 \end{bmatrix} \]

In compact notation, this is written as a \(3 \times 4\) matrix:

\[ \begin{bmatrix} 2 & 1 & 0 & 30 \ 1 & 0 & 4 & 20 \ 0 & 1 & 2 & 10 \end{bmatrix} \]

The closest option from your provided responses is:

\[ \begin{bmatrix} 2 & 1 & 0 & 30 \ 1 & 0 & 4 & 20 \ 0 & 1 & 2 & 10 \end{bmatrix} \]

This corresponds to:

\[ \text{Response: } \left[\begin{array}{cccc} 2 & 1 & 0 & 30 \ 1 & 0 & 4 & 20 \ 0 & 1 & 2 & 10 \end{array}\right] \]

Therefore, the correct option among your responses is the second one:

\[ \begin{bmatrix} 2 & 1 & 0 & 30 \ 1 & 0 & 4 & 20 \ 0 & 1 & 2 & 10 \end{bmatrix} \] (as described in the appropriate response).