Question

Solve each equation.

$\left[\begin{array}{c}x+3y\\ 3x+y\end{array}\right]\mathbf{=}\left[\begin{array}{c}-13\\ 1\end{array}\right]$

Short answer

The solution is $\left(2,-5\right)$.

Step-by-step solution

– Description of step.

As the matrices are equal, the corresponding elements are equal. Write the two linear equations to show this equality.

$\begin{array}{c}x+3y=-13\\ 3x+y=1\end{array}$

– Description of step.

Multiply $3x+y=1$ by 3 and subtract $x+3y=-13$from the obtained equation.

$\begin{array}{c}9x+3y-\left(x+3y\right)=3-\left(-13\right)\\ 9x+3y-x-3y=16\\ 8x=16\end{array}$

– Description of step.

Divide each side of $8x=16$ by 8 to find the value of $x$.

$\begin{array}{c}\frac{8x}{8}=\frac{16}{8}\\ x=2\end{array}$

– Description of step.

Substitute 2 for $x$into $3x+y=1$and simplify to find the value of $y$.

$\begin{array}{c}3\left(2\right)+y=1\\ 6+y=1\end{array}$

– Description of step.

Subtract 6 from both the sides of $6+y=1$ and simplify.

$\begin{array}{c}6+y-6=1-6\\ y=-5\end{array}$

Therefore, the solution is $\left(2,-5\right)$.