# How do you solve #2x - 6y = 6# and #-4x + 12y = -18#?

##### 1 Answer

You can do it either with **substitution** or **elimination**, and you should find that there are *zero solutions*.

**SUBSTITUTION**

#2x - 6y = 6# #color(white)(aaaaaaaaaaa)# (1)

#-4x + 12y = -18# #color(white)(aaaaa)# (2)

Here, you would solve for one variable in terms of the other, and plug in to the unused equation. Let's look at **(1)**.

#2x - 6y = 6#

#=> x = (6 + 6y)/2 = 3 + 3y#

So, you can plug in **(2)** to get:

#-4(3 + 3y) + 12y = -18#

#= -12 cancel(- 12y + 12y) = -18#

**ELIMINATION**

Just so you know how to do it, scale **(1)** and see what you get when you add these two equations together.

#2(2x - 6y = 6)#

#-4x + 12y = -18#

#"-------------------------"#

#0color(white)(aaaaaaaa) = -6#

Again, invalid answer. So let's show why this system has no answers.

**PARALLEL LINES NEVER INTERSECT**

If you compare these equations, you should recognize that if you scale both sides of **(1)** by

#-4x + 12y = -12# #color(white)(aaaaa)# (1)

#-4x + 12y = -18# #color(white)(aaaaa)# (2)

Since these equations are identical except for the constant they are equal to (*never* intersect.

**In other words, this has zero solutions.**