The Substitution Method at a Glance

Doesn't the word 'substitute' conjure up some sort of image of a teacher with pointy eyeglasses and a ruler?

Hmm…disregard that, we're talking the substitution method, not substitute teacher. But the substitution part does mean the same thing as the substitute in substitute teacher—replacing something with another thing.

Sample Problem

Use the substitution method to solve this system of equations.

y = -x + 3

2x + y = 4

Our first question is, why can't we be friends? Next we ask ourselves, what do we want to substitute? Well, the first equation has been conveniently solved for y already. At any spot where there's a y, we can substitute in (-x + 3). And we know where to stick that.

"(-x + 3) can't we be friends, (-x + 3) can't we be friends?" Sing along now.

We suppose it could also go in the second equation too.

2x + (-x + 3) = 4

Well, well. That substitution made y disappear, which will let us solve for x. Usually when things disappear, either a magician or large sums of money are involved.

x + 3 = 4

x = 1

Now we can plug 1 in for x in either of the original equations and solve for y. We'll put it in y = -x + 3.

y = -1 + 3 = 2

We now have a solution to the system of linear equations: (1, 2). That's where the two lines meet. Or is it? Maybe we should double-check.

y = -x + 3

2 = -1 + 3

Our first equation checks out. The left- and right-hand sides agree.

2x + y = 4

2(1) + 2 = 4

Same here. Both sides of both equations agree, so we're in the clear. We're still wondering why (-x + 3) doesn't want to be our friend, though. And (-x + 3), why won't you return our calls? Was it the surprise water balloon? We already said we were sorry.

The Substitution Method

Now that we've done a problem using it, here is the substitution method laid out. Believe us, it's much less scary than that image of a substitute teacher.

Step 1: Isolate one of the variables in one equation. Make sure the variable knows it's not personal, it's just algebra. We don't like to hurt any variables' feelings.

Step 2: Substitute the expression from Step 1 into the other equation. Just cram the whole thing right in there. We should now have an equation with only one variable. Solve for it.

Step 3: Take the answer from Step 2 and substitute it back into either one of the original equations. Solve for the other variable.

Step 4: Check the solution in both of the original equations. Mistakes will be caught, quarantined, and given a good scrub down at this stage.

Optional Step 5: Once everything checks out, bask in the warm glow of algebraic achievement.

That's a lot of words to explain what we're doing, but we don't really need all of them. Let's toss out, say, all but one word for each step: Isolate, Substitute, Solve, Check, Bask. See, it still makes sense.

Sample Problem

Solve using the substitution method.

-2x + y = -1

-x + y = -3

Which variable will we stick in the isolation chamber? That y in the second equation looks like it could use some alone time.

y = x – 3

Now we can substitute (x – 3) in for y in the first equation. We'd substitute it somewhere else, but that joke is already old.

-2x + (x – 3) = -1

-x – 3 = -1

-x = 2

x = -2

We're halfway to a solution to this little problem. All we need is the cooperation of one of our initial equations. Sorry, -2x + y = -1, you've been volunteered. "Conscripted" would be the better word, we suppose.

-2(-2) + y = -1

4 + y = -1

y = -5

Looks like (-2, -5) is our solution. We're so close to basking in our achievement, but we can't until we check our answer. It would be awful if we basked with the wrong answer. Just the absolute worst.

-2x + y = -1

-2(-2) – 5 = -1

4 – 5 = -1

This is good. We're already at half bask; we can't help ourselves.

-x + y = -3

-(-2) – 5 = -3

2 – 5 = -3

Ah, there we go. The achievement washes over us. Excuse us for a moment; we're going to do our best "lizard sitting on a hot rock" impression.

Sample Problem

Solve using substitution method.

-x = 3y + 6

6y + 2x = -12

We haven't given x any time in the isolation chamber yet, so let's get it alone in the first equation.

-x = 3y + 6

x = -3y – 6

We now know x's dirty little secret. Don't worry, x, that isn't anything to be ashamed of. In fact, we'd like to introduce you to the x in the other equation; you two have a lot in common.

6y + 2(-3y – 6) = -12

6y – 6y – 12 = -12

0 = 0

Well, that's different. When all of our variables cancel out and we're left with a true statement like 0 = 0, that means that the lines are actually the same line. They lie on top of each other, so every point on one line is also a solution to the other line. That's an infinite number of solutions. We couldn't count that high, even if you gave us a week.