Question

a) If `x - 2y = -8`, and `3x - 6y = -12`, what are the values of `x` and `y`? b) How do you prove your answer from (a) graphically?

Answer 1

a). The first step in solving by substitution is always solving for one of the variables. Since x in the second equation has coefficient `1`, we'll choose this variable to isolate.

`x - 2y = -8 -> x = 2y - 8`

We now substitute this into the first equation.

`3(2y - 8) - 6y = -12`

`6y - 24 - 6y = -12`

`0y = 12`

This is true for no real value of `y`, therefore this system has no real solution.

b). Let's do a little bit of work with the first equation.

`3x - 6y = -12`

We factor out a `3`.

`3(x - 2y) = -12`

Divide both sides by `3`

`x - 2y = -4`

We get an equation that is identical to the second on the left-hand side, but different on the right-hand side. What does this mean?

Suppose we were to graph both lines. We would first convert to slope-intercept form.

`x - 2y = -4 -> -2y = -4 - x ->y = 1/2x + 2`

For the second equation:

`x - 2y = -8 -> -2y = -8 - x -> y= 1/2x + 4`

These lines have equal slopes but different y-intercepts. This means that these are parallel lines, which is graphical proof that they will never intersect.

Hopefully this helps!