How do i solve systems of equations by elimination?

First make sure that one of the variable has the same coefficient in each equations. 

For example lets use:
y=x+3
2y=3x-1

Since neither the y's or x's are the same, we must multiply the equations to make them the same. I am going to multiply the first equation by 2 so I can eliminate the y's.

So we get:
2y=2x+3
2y=3x-1

Then we subtract the second equation from the first to eliminate the y's. So we end up with 0=-x+4 and now we solve for x
0=-x+4
x=4 

Now we plug in our x into either original to get the y-value. I am using the first
y=x+3
y=4+3
So the x=4 and y=7

Hope this helps!