# How do you solve y = 2/3x - 7 and 2x - 3y = 21 by graphing?

May 7, 2018

Graphically the two equations are just different representations of the same graph, and they, therefore, do not have one definite answer.

#### Explanation:

Normally you would treat the two equations as functions and draw a graph of each. Where they intersect you will find your solution since the solution has to fulfill both equations at the same time.

Here, though, you will find that the two lines overlap.
Both are linear functions, and if you set $y = 0$, you will find that $x = 10 \frac{1}{2}$ solves both equations, i.e. $\left(10 \frac{1}{2} , 0\right)$ is lying on both graphs.
Similarly $x = 0$ gives $y = - 7$, so $\left(0 , - 7\right)$ lies on both graphs.

The two equations, therefore are just two different representations of the same line:
graph{2x-3y=21 [-22.8, 22.8, -11.4, 11.42]}

Algebraically we can see it this way:

Take $2 x - 3 y = 21$
$\left(\frac{2}{3}\right) x - y = \frac{21}{3} = 7$ (Divide each term with 3)
$y = \left(\frac{2}{3}\right) x - 7$ (subtract y+7 from both sides)

We see that the second equation is converted to the first one, so they are identical and, therefore, represent the same equation.