Which option accurately re-creates the last row of the augmented matrix for the following three-variable system of equations?

Question

⎧⎩⎨⎪⎪6x−y+4z=612x+y−2z=12−24x+y+3z=−24

(1 point)
Responses

[2413–24]
An augmented matrix is shown. Start 1 By 4 Matrix 1st Row 1st Column 24 2nd Column 1 3rd Column 3 4th Column negative 24 End Matrix. A vertical line separates the third and fourth columns of the matrix.

[241324]
An augmented matrix is shown. Start 1 By 4 Matrix 1st Row 1st Column 24 2nd Column 1 3rd Column 3 4th Column 24 End Matrix. A vertical line separates the third and fourth columns of the matrix.

[–2413–24]
An augmented matrix is shown. Start 1 By 4 Matrix 1st Row 1st Column negative 24 2nd Column 1 3rd Column 3 4th Column negative 24 End Matrix. A vertical line separates the third and fourth columns of the matrix.

[–241324]
An augmented matrix is shown. Start 1 By 4 Matrix 1st Row 1st Column negative 24 2nd Column 1 3rd Column 3 4th Column 24 End Matrix. A vertical line separates the third and fourth columns of the matrix.

Answer 1

To find the last row of the augmented matrix corresponding to the given system of equations, we need to analyze the equations provided:

1. \(6x - y + 4z = 6\)
2. \(12x + y - 2z = 12\)
3. \(-24x + y + 3z = -24\)

In matrix form, the augmented matrix will look like this:

\[ \begin{bmatrix} 6 & -1 & 4 & | & 6 \ 12 & 1 & -2 & | & 12 \ -24 & 1 & 3 & | & -24 \end{bmatrix} \]

The last row of the augmented matrix corresponds to the third equation. Therefore, the last row will be:

\[ [-24 \quad 1 \quad 3 \quad | \quad -24] \]

This representation can be put into the following options:

• Option A: \([-24 \quad 1 \quad 3 \quad -24]\)

Hence, the correct option that accurately re-creates the last row of the augmented matrix is: [–2413–24].