In this section we're not going to cover anything new; I'm just going to remind you of some things you've already known from your previous studies in algebra, and show you how these things apply to solving systems of equations. In the previous section I gave you several problems which were nicely formatted with the x term first, the y term second, and the constant on the other side of the equation. That made it easy to quickly see what you need to do to make a variable cancel. Will problems always be that nicely formatted? Of course not! When have math teachers ever been that nice?

Example #1

x + y = 7
2y - x = 8

Here it's tempting to say, "Oh! If I just add the equations, the second variable will cancel out!" But wait a minute, what is the second variable? In the first equation the second variable is y, and in the second equation it's x! You know what that means, right? They're not like terms, and nothing will cancel. So it's always a good idea to rearrange things to get the variables lined up. It'll make it easier for you to see what you need to do. So we'll rewrite the second equation with the x term first and the y term second:

x + y = 7
-x + 2y = 8

Now it's easy to see that if we add the equations, the x variable will cancel:

 x +  y =  7
-x + 2y =  8
____________
     3y = 15
      y =  5 

Then, of course, we plug that value into either equation and find that x = 2, and then we can check the answer by plugging that into the other equation.

Example #2

x + 7 = y - 2
x + y = 3

In this system, the first equation is definitely not in a form we're used to seeing. So let's use the algebra rules we know for rearranging equations to get it in the form we want:

x + 7 = y - 2
  - 7     - 7 (subtract 7 from both sides)
_____________
x     = y - 9
  - y  -y     (subtract y from both sides)
_____________
x - y = -9 

Oh, this is much better! Now we have:

x - y = -9
x + y = 3

I don't need to continue working this one out; you know how to take it from here!

Example #3

2(3x + 3) = y
3(y - 10) = 2x + 20

Wow! Those are ugly! But they're also no problem, because you know the distributive property, so you can easily rearrange these:

2(3x + 3) = y
6x + 6    = y       (distributive property)
    -6         -6   (subtract 6 from both sides)
_________________
6x        = y - 6
    -y     -y       (subtract y from both sides)
_________________

6x - y         = -6

Now rarrange the other equation:

3(y - 10) = 2x + 20
3y - 30   = 2x + 20 (distribute)
   + 30        + 30 (add 30)
___________________
3y        = 2x + 50
    - 2x   -2x      (subtract 2x)
___________________
3y - 2x   = 50
-2x + 3y  = 50      (rearrange so the order of x and y match)

You may finish solving this as an exercise.