Solve using augmented matrices. -3x + y = -15, 7x + y = 15.

Steps to Solve

1. Write the Augmented Matrix We start by converting the system of equations into an augmented matrix. The matrix represents the coefficients of the variables and the constants from the equations:

$$ \begin{bmatrix} -3 & 1 & | & -15 \ 7 & 1 & | & 15 \end{bmatrix} $$

2. Row Operations to Form Row Echelon Form Next, we will perform row operations to simplify the augmented matrix. We want to eliminate one variable from one equation.

First, we can multiply the first row by 7 and the second row by 3 to align the coefficients of $x$:

Row 1: $7 \cdot R_1 \Rightarrow \begin{bmatrix} -21 & 7 & | & -105 \end{bmatrix}$

Row 2: $3 \cdot R_2 \Rightarrow \begin{bmatrix} 21 & 3 & | & 45 \end{bmatrix}$

This gives us:

$$ \begin{bmatrix} -21 & 7 & | & -105 \ 21 & 3 & | & 45 \end{bmatrix} $$

3. Add Rows to Eliminate One Variable Now, we can add Row 1 and Row 2 to eliminate $x$:

$$ R_1 + R_2 \Rightarrow \begin{bmatrix} 0 & 10 & | & -60 \end{bmatrix} $$

So, the new matrix becomes:

$$ \begin{bmatrix} -21 & 7 & | & -105 \ 0 & 10 & | & -60 \end{bmatrix} $$

4. Solve for y Now we solve the second row for $y$:

From $0y + 10y = -60$, we find

$$ y = \frac{-60}{10} = -6 $$

5. Substitute y Back to Find x Now, substitute $y = -6$ into one of the original equations to find $x$. We'll use the first equation:

$$ -3x + (-6) = -15 $$

Simplifying gives:

$$ -3x - 6 = -15 $$

Adding 6 to both sides:

$$ -3x = -15 + 6 $$ $$ -3x = -9 $$

Solving for $x$ gives:

$$ x = \frac{-9}{-3} = 3 $$