In the following exercises, solve the systems of equations by elimination.

$\left\{\begin{array}{l}5x+2y=1\\ -5x-4y=-7\end{array}\right.$

We are given a system of equations.

$\left\{\begin{array}{l}5x+2y=1\\ -5x-4y=-7\end{array}\right.$

We need to solve the system using the method of elimination.

The two equations in the system are written in standard form and the coefficients of $x$are already opposites. So we add the two equations

$\begin{array}{rcl}5x+2y+(-5x-4y)& =& 1+(-7)\\ 5x+2y-5x-4y& =& 1-7\\ -2y& =& -6\\ y& =& \frac{-6}{-2}\\ y& =& 3\end{array}$

Substitute $3$for $y$in the first equation.

$\begin{array}{rcl}5x+2\left(3\right)& =& 1\\ 5x+6& =& 1\\ 5x& =& 1-6\\ 5x& =& -5\\ x& =& \frac{-5}{5}\\ x& =& -1\end{array}$

So the solution of the system is$(-1,3)$.

Substitute $-1$for $x$and $3$for $y$in the first equation.

$\begin{array}{rcl}5x+2y& =& 1\\ 5(-1)+2\left(3\right)& =& 1\\ -5+6& =& 1\\ 1& =& 1\end{array}$

As the statement is correct, so the solution satisfies the first equation.

Substitute $-1$for $x$and $3$for $y$in the second equation.

$\begin{array}{rcl}-5x-4y& =& -7\\ -5(-1)-4\left(3\right)& =& -7\\ 5-12& =& -7\\ -7& =& -7\end{array}$

As the statement is correct, so the solution satisfies the second equation.