A system of equations is simply a collection of equations that share a relationship. Here, you’ll learn how to solve two equations with two unknowns—two variables. You’ll learn three methods to solve systems of equations: Graphing, the substitution method, and the elimination method. Let’s start with graphing.

Rule

- 1.
- Solve both equations for $y$. This means that you should have $y$ by itself on one side of both expressions. It looks something like this: $y=ax+b$.
- 2.
- Your expressions are now linear functions. Draw both of them in the same coordinate system.
- 3.
- Find the point of intersection and write down the corresponding values from the $x$-axis and the $y$-axis.

Example 1

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

First you solve (1) with respect to $y$:

$$y=2x+2$$ |

Then you solve (2) with respect to $y$: $$\begin{array}{llll}\hfill 4y+4x& =20\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 4y& =-4x+20\phantom{\rule{1em}{0ex}}|\xf74\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill y& =-x+5\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$

You may now plot the lines $y=2x+2$ and $y=-x+5$:

The point of intersection gives you $x=1$ and $y=4$.

ANSWER: $(x,y)=(1,4)$

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Systems of Equations (Substitution)