Solve each system of equations by graphing

$\begin{array}{l}\text{4}x-2y=22\\ 6x+9y=-3\end{array}$

Equation of line in slope intercept form is expressed below.

$y=mx+c$

Where m is the slope and c is the intercept of y-axis.

Consider the first equation $4x-2y=22$.

Rewrite the equation in form of slope-intercept form.

Subtract both sides by $4x$

$\begin{array}{l}4x-2y-4x=22-4x\\ -2y=22-4x\end{array}$

Next, divide both sides by $-2$and rearrange the terms.

$\begin{array}{l}\frac{-2y}{-2}=\frac{22-4x}{-2}\\ y=\frac{22}{-2}+\frac{4x}{2}\\ y=2x-11\end{array}$

Now, the equation is in the form $y=mx+c$. Here slope m of the line is $2$ and intercept of y-axis c is $11$.

Now, consider the second equation $6x+9y=-3$.

Rewrite the equation in form of slope-intercept form.

Subtract both sides by $6x$

$\begin{array}{l}6x+9y-6x=-3-6x\\ 9y=-3-6x\end{array}$

Next, divide both sides by 9 and rearrange the terms.

$\begin{array}{l}\frac{9y}{9}=\frac{-3-6x}{9}\\ y=-\frac{3}{9}-\frac{6x}{9}\\ y=-\frac{2}{3}x-\frac{1}{3}\end{array}$

Now, the equation is in the form $y=mx+c$. Here slope m of the line is $-\frac{2}{3}$ and intercept of y-axis c is $-\frac{1}{3}$.

Plot the equations on the same plane and the point where both the equations intersect is the solution of the system of the equations.

The red line denotes the equation $4x-2y=22$ and blue line denotes the equation $6x+9y=-3$.

Therefore, the point of intersection is$\left(4,-3\right)$.

The point of intersection is solution of system of equations if the point satisfies both the equation.

Substitute the point $\left(4,-3\right)$in the equation$4x-2y=22$.

Substitute x as 4 and y as $-3$and check whether right hand side is equal to left hand side of the equation

$\begin{array}{l}4\left(4\right)-2(-3)=22\\ 16+6=22\\ 22=22\end{array}$

Since, this is true so the point $\left(4,-3\right)$satisfy the equation $4x-2y=22$.

Substitute the point $\left(4,-3\right)$in the equation $6x+9y=-3$.

Substitute x as 4 and y as $-3$and check whether right hand side is equal to left hand side of the equation.

$\begin{array}{l}6\left(4\right)+9(-3)=-3\\ 24-27=-3\\ -3=-3\end{array}$

Since, this is true so the point $\left(4,-3\right)$satisfy the equation$6x+9y=-3$.

Hence, the solution of the system of equations is $\left(4,-3\right)$.