# How do you solve by substitution y = 3x – 4 and 9x – 3y = 14?

Jun 12, 2015

These equations represent parallel lines so there is no solution.

#### Explanation:

We can attempt a solution as follows:
[1]$\textcolor{w h i t e}{\text{XXXX}}$$y = 3 x - 4$
[2]$\textcolor{w h i t e}{\text{XXXX}}$$9 x - 3 y = 14$

Using [1], substitute $\left(3 x - 4\right)$ for $y$ in [2]
[3]$\textcolor{w h i t e}{\text{XXXX}}$$9 x - 3 \left(3 x - 4\right) = 14$
Simplify
[4]$\textcolor{w h i t e}{\text{XXXX}}$$4 = 14$
Obviously impossible!

So, what went wrong?
If we take [1] and rearrange it into standard form:
[5]$\textcolor{w h i t e}{\text{XXXX}}$$3 x - y = 4$
then multiplying by 3
[6]$\textcolor{w h i t e}{\text{XXXX}}$$9 x - 3 y = 12$

What are we to make of the combination of [2] and [6]?
$\textcolor{w h i t e}{\text{XXXX}}$We have $\left(9 x - 3 y\right)$ equal to 2 different values!

If we graph these two equations, we would see that they form a pair of parallel lines. (Go ahead; try it. I'll wait).

If two lines are parallel then they do not intersect and there is no point where they are equal.