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

Feb 25, 2016

There is no single solution to the system of equations.

#### Explanation:

Given
[1]$\textcolor{w h i t e}{\text{XXX}} y = \frac{2}{3} x - 7$
[2]$\textcolor{w h i t e}{\text{XXX}} 2 x - 3 y = 21$

Suppose we attempt this one "by substitution"
Substitute $\frac{2}{3} x - 7$ (from equation [1]) for $y$ in equation [2]
[3]$\textcolor{w h i t e}{\text{XXX}} 2 x - 3 \left(\frac{2}{3} x - 7\right) = 21$

Simplifying
[4]$\textcolor{w h i t e}{\text{XXX}} 2 x - 2 x + 21 = 21$
[5]$\textcolor{w h i t e}{\text{XXX}} 21 = 21$
Whoa! we ended up with a true equation but no variable solution!

What does this mean?
It means that the original equations were not independent; they are just rearranged versions of each other.

In terms of a graph the two equations represent the same line and therefore do not intersect to give a single solution.