The Graphing Method at a Glance

When we're asked to solve a system of linear equations for x and y, we're really being asked to find the (x, y) point where these lines meet. Do they meet at (1, 1)? Pittsburgh? The fourth moon of Neptune? How on Earth (or Neptune) do we find the answer?

We could guess (x, y) values—easy if the answer is (1, 1), but tough for . Besides, we already said we were kidding about guesswork. Let's try graphing instead.

A Plot for Successful Graphing

First, find some graph paper—we don't want anyone scratching a grid into the kitchen table. Then graph both lines on the same coordinate grid. When they're both in slope-intercept form, we can do this by plotting the intercepts and then using the slope to find another point. Don't try to intercept the plots. Let the spies do that. It's their job.

If we graph two equations and both of them happen to be the exact same line, then that system has an infinite number of solutions. That's because one line is exactly on top of the other and both lines will extend, in the words of Buzz Lightyear, "to infinity and beyond." Except there's really no "beyond infinity." You get the idea, though.

On the other hand, if we graph two equations and they happen to be parallel to each other, there is no intersection. They never meet up. That is the definition of parallel lines, after all. We can plug in numbers for three months and we won't find an intersection, and we'll waste our entire summer.

Let's not completely diss plugging in numbers, though, because it's a useful method for checking our answers. To see if a point is a solution for a system of equations, simply plug the point into each equation individually and check. The point must satisfy both equations. No exceptions allowed. We're very picky.

Sample Problem

Solve the following system of equations.

y = 2x – 2

First, plot both lines on the same coordinate grid.

This seems clear enough: these lines meet at the point (2, 2). However, we have to double-check that it's actually a solution for the system. There's no shortcut to success, after all.

Did you hear about Billy? He tried to take a shortcut to success, but he took a shortcut into the Forbidden Zone instead. True story. Let's plug (2, 2) into our equations to see if it satisfies them both.

2 = 2(2) – 2

2 = 4 – 2

2 = 2

Why yes, 2 always does equal itself. Well done.

2 = -3 + 5

2 = 2

We can now confirm that 2 continues to equal itself even now, ten seconds into the future. That means that (2, 2) is a solution to this system of equations.

Sample Problem

Solve the system of linear equations:

3x + 3y = 9

x + y = 3

Maybe by graphing? Let's do it by graphing.

Uh, didn't we start with two equations? Maybe one of them forgot to take the left turn at Albuquerque? Or maybe both of them describe the same line? We'll check that plugging some points into both equations, to see if we get the same results. If that doesn't pan out, we'll ask a wascally wabbit for directions.

Peeking at the graph, (0, 3), (1, 2), and (3, 0) all look to be on the line:

3(0) + 3(3) = 9

0 + 3 = 3

Check.

3(1) + 3(2) = 3 + 6 = 9

1 + 2 = 3

Double-check.

3(3) + 3(0) = 9

3 + 0 = 3

T-t-t-triple-check. We can confidently say that there are infinitely many solutions to this system of equations. We could keep plugging numbers in, but what's the point? They're the same line. Guess we won't need these carrots after all.