Intermediate Algebra, Chapter 4, Review Exercises, Section Review Exercises, Problem 26

Use row operations to solve the system $
\begin{equation}
\begin{aligned}

2x + 5y =& -4 \\
4x - y =& 14

\end{aligned}
\end{equation}
$
.

Augmented Matrix

$\displaystyle \left[
\begin{array}{cc|c}
2 & 5 & -4 \\
4 & -1 & 14
\end{array}
\right]$

$\displaystyle \frac{1}{2} R_1$

$\displaystyle \left[
\begin{array}{ccc}
1 & \displaystyle \frac{5}{2} & -2 \\
4 & -1 & 14
\end{array}
\right]$

$\displaystyle R_2 - 4R_1 \to R_2$

$\displaystyle \left[
\begin{array}{ccc}
1 & \displaystyle \frac{5}{2} & -2 \\
0 & -11 & 22
\end{array}
\right]$

$\displaystyle - \frac{1}{11} R_2$

$\displaystyle \left[
\begin{array}{ccc}
1 & \displaystyle \frac{5}{2} & -2 \\
0 & 1 & -2
\end{array}
\right]$

This augmented matrix leads to the system of equations.


$
\begin{equation}
\begin{aligned}

x + \frac{5}{2}y =& -2
\\
\\
y =& -2

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

x + \frac{5}{2} (-2) =& -2
&& \text{Substitute $y = -2$ in equation 1}
\\
x - 5 =& -2
&& \text{Multiply}
\\
x =& 3
&& \text{Add each side by $5$}

\end{aligned}
\end{equation}
$


The solution set of the system $\{ (3,-2) \}$.