Question

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

Answer 1

These equations represent parallel lines so there is no solution.

Explanation:

We can attempt a solution as follows:
[1]`color(white)("XXXX")` `y = 3x-4`
[2]`color(white)("XXXX")` `9x-3y = 14`

Using [1], substitute `(3x-4)` for `y` in [2]
[3]`color(white)("XXXX")` `9x-3(3x-4) = 14`
Simplify
[4]`color(white)("XXXX")` `4 = 14`
Obviously impossible!

So, what went wrong?
If we take [1] and rearrange it into standard form:
[5]`color(white)("XXXX")` `3x-y = 4`
then multiplying by 3
[6]`color(white)("XXXX")` `9x-3y =12`

What are we to make of the combination of [2] and [6]?
`color(white)("XXXX")`We have `(9x-3y)` 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.