 # How to Graphically Solve Systems of Equations

Here you will learn how to use graphic solving to solve systems of equations through a set of instructions that will always get you to the finish line!

Rule

### SolvingaSystemofEquationsGraphically

1.
Solve both equations for $y$. That means you need $y$ on its own on the left side in both expressions. It should look like this:
 $y=ax+b.$
2.
Your expressions are now linear functions. Draw them in the same coordinate system.
3.
Find where the two graphs intersect, taking note of one number from the $x$ axis and one from the $y$ axis—the x- and y- coordinates.

Example 1

Solve the system of equations: $\begin{array}{lll}\hfill y-2x& =2\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\phantom{\rule{0.33em}{0ex}}\\ \hfill 4y+4x& =20\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\phantom{\rule{0.33em}{0ex}}\end{array}$

1.
First solve (1) with respect to $y$:
 $y=2x+2.$

Then you solve (2) with respect to $y$: $\begin{array}{llll}\hfill 4y+4x& =20\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 4y& =-4x+20\phantom{\rule{1em}{0ex}}|÷4\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill y& =-x+5\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

2.
Draw the graphs of the expressions for $y$ from the previous point: 3.
Find the intersection at $x=1$ and $y=4$, and write the answer like this:
 $\text{Answer:}\left(x,y\right)=\left(1,4\right)$

Recall that there are three different kinds of solutions. There are solutions where the graphs do not intersect, solutions where the graphs intersect at one point, and solutions where the graphs are on top of each other and intersect at every point.

If you’re ever unsure of the solution you find through algebra, you can always get some perspective by drawing the graphs of the equations. I always feel like it helps to have an image of how everything connects together.

With training you will find that you can sketch these in your mind, which is very useful!