Solve using augmented matrices: -2x + 2y = -8; y = -10.

Steps to Solve

1. Set up the augmented matrix

Convert the system of equations into an augmented matrix. The given equations are:

[ -2x + 2y = -8 ]

[ y = -10 ]

This can be represented in augmented matrix form as:

$$ \begin{bmatrix} -2 & 2 & | & -8 \ 0 & 1 & | & -10 \end{bmatrix} $$

2. Use row operations to simplify the matrix

The goal is to simplify the augmented matrix to row-echelon form. Begin by interchange the rows if necessary and scaling.

In this case, we can use the second row to eliminate (y) from the first row by replacing the first row with the operation:

[ R_1 \leftarrow R_1 + 10R_2 ]

This results in:

$$ \begin{bmatrix} -2 & 0 & | & -8 + 10(-10) \ 0 & 1 & | & -10 \end{bmatrix} $$

Simplifying the first row gives:

$$ \begin{bmatrix} -2 & 0 & | & 92 \ 0 & 1 & | & -10 \end{bmatrix} $$

3. Solve for (x) and (y)

From the second row, we can directly solve for (y):

[ y = -10 ]

Now substitute (y) back into the first equation to solve for (x):

From the first row equation, we have:

[ -2x = 92 ]

Divide both sides by -2:

[ x = -46 ]

4. Write the final solution

The solution to the system of equations is:

[ x = -46, \quad y = -10 ]