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

Steps to Solve

1. Write the Augmented Matrix

First, we convert the system of equations into an augmented matrix. The system is:

$$ \begin{align*} -2x + 2y &= -8 \ y &= -10 \end{align*} $$

This can be represented in augmented matrix form as:

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

2. Apply Row Operations

Next, we'll perform row operations to simplify the matrix. We can start by isolating the variable. Let's focus on making the first element of the first row a leading 1.

To do this, we'll multiply the first row by $-\frac{1}{2}$:

$$ R_1 \rightarrow -\frac{1}{2} R_1 $$

This gives us:

$$ \begin{bmatrix} 1 & -1 & | & 4 \ 0 & 1 & | & -10 \end{bmatrix} $$

3. Eliminate the Second Row Variable

Next, we need to eliminate the $y$ term from the first row. We can do this by replacing the first row with:

$$ R_1 \rightarrow R_1 + R_2 $$

This results in:

$$ \begin{bmatrix} 1 & 0 & | & -6 \ 0 & 1 & | & -10 \end{bmatrix} $$

4. Interpret the Augmented Matrix

Now that we have the matrix in reduced row echelon form, we interpret it:

The first row indicates:

$$ x = -6 $$

The second row indicates:

$$ y = -10 $$