This method for solving systems of equations goes by many names: the addition method, elimination method, or solving by "linear combination." Doesn't that sound like fun?
It's not that bad, though. Once you get the hang of it, it's amazingly quick. It's quicker than your little brother chasing the ice-cream truck.
Sample Problem
Solve this linear system of equations using the elimination method.
3x – 3y = -1
2x + 3y = 6
The secret to the elimination method is that we can add our two equations together top-to-bottom, and if we can solve the new equation for x or y, it will be part of the solution to the original equations. It's not Houdini-level magic, but it's a neat trick nonetheless.
Looking at our system of equations, we see that we have 3y and -3y. If we add the equations together, the y's will eliminate each other out, leaving us with only x's. The y's will have to sit out until the next round.
(3x + 2x) + (-3y + 3y) = (-1 + 6)
5x = 5
x = 1
All right, y, you can come back into the game. Now, 1, we want you to take x's spot on the field. Together, try to take on 2x + 3y = 6 and score a few points. Maybe find the true meaning of y while you're at it.
2(1) + 3y = 6
3y = 4
Oh, y was a fraction all along. Our solution is 
. We need to do a background check on this.
3x – 3y = -1
3 – 4 = -1
So far, so good.
2x + 3y = 6
2 + 4 = 6
So farther, even better.
The elimination method works great when all of the variables have coefficients attached to them. If those numbers are harder to get off than cling wrap, then just eliminate them. Trying to use substitution would be too much of a hassle. Not that we can stop you from trying.
Sample Problem
Solve this linear system using the elimination method.
x + y = 2
2x + 3y = 4
Even in a case like this, where the substitution is practically done for us, we can still use elimination.
In this case, though, we need to take an extra step first. If we add our equations together now, nothing will be eliminated. That helps no one, except the Advocates for Unsolved Equations.
We need to multiply one or both of our equations to create some additive inverses. Multiplying x + y = 2 by -2 will do the trick. Now our system of equations looks like this:
-2x – 2y = -4
2x + 3y = 4
Let's get eliminating.
(-2x + 2x) + (-2y + 3y) = (-4 + 4)
0 – y = 0
Solving for y is so easy at this point, that y = 0. Oh, we couldn't even wait until we finished that sentence in order to do it.
Now we plug y into one of the initial equations and get our answer. Use the originals, not the ones we multiplied by, just in case we made some mistake before.
x + 0 = 2
x = 2
Our solution is (2, 0). But now we check it, because we're super careful with this kind of stuff.
x + y = 2
2 + 0 = 2
That checks out.
2x + 3y = 4
2(2) + 3(0) = 4
And we're golden. Not literally, of course. Otherwise, we would be constantly fighting off people wanting a piece of us. More than usual, anyway.
Sample Problem
Solve this linear system of equations using the elimination method.
y = x + 7
x – y = 8
We're feeling a little dangerous right now. We're just going to add up each side of these equations right now, as they are. "Oh no, the terms aren't all lined up." Well boo-hoo, we'll sort that out later.
(y) + (x – y) = (x + 7) + (8)
x = x + 15
See? It worked out all right in the end. We eliminated y, didn't we? Now let's get all our x's on the same side and solve.
(x – x) = 15
0 = 15
Oh no, we take it back, we take it back. Did we make a mistake in our hubris? No, the math checks out, again and again and again and again. Yes, we were so freaked out we checked it four times.
Wait, this kind of nonsense is what we get when a system of equations has no solutions. The graphs never cross, so they should be parallel. We can check that.
To graph y = x + 7, we can use the y-intercept and then plug in a value for x to get another point. For the y-intercept, that's b = 7, or (0, 7) since we want to graph it. When x = 1, we have y = 8. This makes the line look like:
To graph x – y = 8, we can find the x- and y-intercepts. For the x-intercept, let y = 0. That means x – 0 = 8, so x = 8. This gives us the point (8, 0) where this line crosses the x-axis. For the y intercept, x = 0. That gives us 0 – y = 8, so y = -8.
We almost have enough intercepts to film a movie. Interception: You have to graph deeper.
Just as we said, our system has no solutions because the graphs are parallel. It's nice to know we didn't goof anything up.
Sample Problem
Solve using the elimination method.
3x – 5y = 6
6x – 10y = 12
We see immediately that multiplying the first equation by -2 will eliminate x. Go go go for it.
-2(3x – 5y) = -2(6)
-6x + 10y = -12
Let's just add our equations together and—huwawa?
6x – 10y = 12
-6x + 10y = -12
0 = 0
Surprise! Our lines are secret twins. See, 0 will always be equal to 0. Therefore, there are infinitely many solutions. The lines sit on top of each other, so any point found on one line has to be on the other one too.
Elimination Summary
Let's review what to do when we want to use the elimination method. None of the steps require anyone to sleep with the fishes.
Step 1: Multiply one or both of the equations by a constant so that the coefficients for the same variable in both equations only differ by sign. Like STOP and YIELD? No, silly. Like plus and minus.
Step 2: Add together the revised equations for Step 1. Combining like terms will eliminate one of the variables. How about that—one variable just drops out. (Don't feel bad, we'll bring it back in the next step.) Now solve for the remaining variable.
Step 3: Substitute the value obtained in Step 2 into either of the original equations and solve for the other variable. Follow with more achievement-based glow-basking. You've earned it at that point.