The Substitution Method

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There are two methods we can use to find the intersection of two lines: the substitution method and the addition method. Of the two, the substitution method is definitely the easiest as long as you can avoid bringing fractions into the process. You'll see what I mean by this in the examples.

The process for using the substitution method goes like this:

Solve one of the equations for one of its variables. You can do it with either one but the later steps will be a lot easier if you can avoid having any fractions.
Substitute the expression you found in step one into the other equation. (This time it has to be the other equation.)
Solve the equation that you made in step two for its variable.
Substitute the value you found in step three into one of the original equations and solve for the other variable.

Example 1

Find the intersection of the lines y + 3x = 2 and 2y - 2x = 1.

If you look at the first equation, you'll see that the y variable doesn't have a number in front of it. That's a sign that the substitution method is a good option. I'm going to start by taking that equation and solving it for y.

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

This is why there not being a number in front of the y was a good sign. If there had been, when we divided both sides by it, we probably would have introduced a fraction which would make latter steps complicated.

Now I'm going to take 2 - 3x and substitute it for y in the other equation then solve it for x.

$$2(2 - 3x) - 2x = 1$$ $$4 - 6x - 2x = 1$$ $$4 - 8x = 1$$ $$-8x = -3$$ $$x=\frac{3}{8}$$

That's going to be the y value of our intersection. Now, to get the x value, I'm going to take that number and substitute it into the second equation then solve for x.

$$2y - 2\cdot \frac{3}{8} = 1$$ $$2y - \frac{3}{4} = 1$$ $$2y = 1 - \frac{3}{4}$$ $$2y = \frac{1}{4}$$ $$y = \frac{1}{8}$$

So the coordinates of the intersection of our two lines must be $\left(\frac{3}{8}, \frac{1}{8}\right)$.

Example 2

Find the intersection of the lines 3y - 2x = 1 and 4y - 5x = 2.

First, I'll take the first equation and solve it for y.

$$3y-2x=1$$ $$3y = 1 + 2x$$ $$y = \frac{1}{3} + \frac{2}{3}x$$

Now I'll substitute that into the other equation to get one that I can solve for x.

$$4\left(\frac{1}{3} + \frac{2}{3}x\right) - 5x = 2$$ $$\frac{4}{3} + \frac{8}{3}x - 5x = 2$$ $$\frac{4}{3} - \frac{7}{3}x = 2$$ $$-\frac{7}{3}x = 2 - \frac{4}{3} = \frac{2}{3}$$ $$x=-\frac{3}{7} \cdot \frac{2}{3} = -\frac{2}{7}$$

Now, I'll take that value and substitute it into the first equation to get the y-coordinate of the intersection.

$$3y - 2\frac{-2}{7}=1$$ $$3y + \frac{4}{7}=1$$ $$3y = 1 - \frac{4}{7} = \frac{3}{7}$$ $$y = \frac{1}{7}$$

So the coordinates of the intersection of our two lines must be $\left(-\frac{2}{7}, \frac{1}{7}\right)$.

Video Lectures

Lectures

Systems of Linear Equations (2 Variables, Substitution)

Substitution is the most straightforward of the methods for solving systems of equations but it isn't always the easiest. In this lecture, we'll look at the method and talk about when it's the best one to use. (lecture slides)

Two Unreliable Methods

There are two methods, graphing and so-called numeric solutions, that you really should avoid. In this lecture, I'll go over both methods and why you can't rely on them. (lecture slides)

Dyanmic Tutorial - Using the Substitution Method

Directions: This solution has 6 steps. To see a description of each step click on the boxes on the left side below. To see the calculations, click on the corresponding box on the right side. Try working out the solution yourself and use the descriptions if you need a hint and the calculations to check your solution.