Question

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

Answer 1

There is no single solution to the system of equations.

Explanation:

Given
[1]`color(white)("XXX")y=2/3x-7`
[2]`color(white)("XXX")2x-3y=21`

Suppose we attempt this one "by substitution"
Substitute `2/3x-7` (from equation [1]) for `y` in equation [2]
[3]`color(white)("XXX")2x-3(2/3x-7)=21`

Simplifying
[4]`color(white)("XXX")2x-2x+21=21`
[5]`color(white)("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.