How to Graph Linear Equations by Substitution

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

Linear equation graphs are always straight lines, so substitution can be an easy way to draw the graph. In this lesson, we'll explore how to find and use the points you need, so graphing linear equations can be easy!

What is a Linear Equation?

The graph of a linear equation is always a straight line, which makes it really easy to graph if you can only substitute a few values into the formula. It's sort of a 'plug and play' situation, where you plug in the values and see what comes out. In this lesson, we'll practice feeding in values and seeing what the resulting graphs look like.

A linear equation is a relationship between two or more values where the graph is a straight line. Any variables in the equation will have an exponent of 1, which means the exponent won't appear at all. Since the graph is a straight line, you can find just two points, connect them with a straight line, and you've got the graph!

The Slope-Intercept Form

One way you will see linear equations will be in the slope-intercept form, or y = mx + b, where x and y are the variables and m and b are numbers that help define the relationship. For example, if you have the equation y = 2x + 4, that means that for this particular set of x and y pairs, the slope is 2, which means the graph will climb or fall twice as fast as it goes forward. The y-intercept is 4, which means that if x is 0, y will end up being 4 and the graph will cross the y-intercept at y=4.

So let's try a couple of substitutions for the equation. Maybe you want to start with x = 0, since that's a nice easy one.

Step 1: y = 2x + 4 (original equation)

Step 2: y = 2 (0) + 4 (substitute zero for x)

Step 3: y = 4 (simplify and solve)

So we can see that y is 4 when x = 0, so that gives us one point for our graph (0, 4). Now we need another one. Let's pick one a little ways to the right, maybe x = 3.

Step 1: y = 2x + 4 (original equation)

Step 2: y = 2 (3) + 4 (substitute 3 for x)

Step 3: y = 10 (simplify and solve)

There's our second point (3, 10). Okay, let's graph it.

Figure 1
graph image

Take a look at Figure 1. There is an x axis across the middle going left to right and a y axis going up and down in the center. Every fifth grid line is numbered to help us keep track of where we are. The two points we calculated appear as dots on the graph, and then the two dots were connected and the line extended to the edges of our graph. That's all there is to it!

Point-Slope Form

The point-slope form, (y - y1) = m (x - x1), can be a little more confusing, but it's the same idea. For example, let's take the equation (y - 2) = 2 (x + 2). We'll use the same approach. Let's feed it a few values for x and see where y goes. A favorite value for x is 0, so let's start there.

Step 1: (y - 2) = 2 (x + 2) (original equation)

Step 2: (y - 2) = 2 (0 + 2) (substitute 0 for x)

Step 3: (y - 2) = 4 (multiply 2 by the sum of 0 and 2)

Step 4: y = 6 (add 2 to both sides)

One pair is (0, 6). All right, let's give it another value for x, perhaps 5, and see where that takes us.

Step 1: (y - 2) = 2 (x + 2) (original equation)

Step 2: (y - 2) = 2 (5 + 2) (substitute 5 for x)

Step 3: (y - 2) = 14 (multiply 2 by the sum of 5 and 2)

Step 4: y = 16 (add 2 to both sides)

16 is a pretty large number, and we only drew our graph out to about 10 grid lines last time, so we can either change the shape of the graph or pick a smaller number. Just for fun, let's use x = -2 and see what happens.

Step 1: (y - 2) = 2 (x + 2) (original equation)

Step 2: (y - 2) = 2 (-2 + 2) (substitute -2 for x)

Step 3: (y - 2) = 0 (multiply 2 by the sum of -2 and 2)

Step 4: y = 2 (add 2 to both sides)

Well, that will certainly fit on our graph, so let's use (0, 6) and (-2, 2). Take a look at Figure 2.

Figure 2
linear graph

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