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

1 Answer

Feb 25, 2016

There is no single solution to the system of equations.

Explanation:

Given
[1] $XXX y = \frac{2}{3} x - 7$ $color(white)("XXX")y=2/3x-7$
[2] $XXX 2 x - 3 y = 21$ $color(white)("XXX")2x-3y=21$

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

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

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