Beginning Algebra
Tutorial 21: Graphing Linear Equations

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Learning Objectives

After completing this tutorial, you should be able to:

Recognize when an equation in two variables is a linear equation.
Graph a linear equation.

Introduction

In Tutorial 20: The Rectangular Coordinate System, we went over the basics of the rectangular coordinate system. In this tutorial we will be adding on to this by looking at graphing linear equations by plotting points that are solutions. Basically, when you graph, you plot solutions and connect the dots to get your graph. Specifically, when you graph linear equations, you will end up with a straight line. Let's see what you can do with these linear equations.

Tutorial

Linear Equation in
Two Variables

Standard Form:

Ax + By = C

A linear equation in two variables is an equation that can be written in the form
Ax + By = C, where A and B are not both 0.

This form is called the standard form of a linear equation.

Example 1: Determine whether the equation y = 5x - 3 is linear or not.

If we subtract 5x from both sides, then we can write the given equation as -5x + y = -3.

Since we can write it in the standard form, Ax + By = C, then we have a linear equation.

If we were to graph this equation, we would end up with a graph of a straight line.

Example 2: Determine whether the equation is linear

or not.

If we subtract the x squared from both sides, we would end up with

. Is this a linear equation? Note how we have an x squared as opposed to x to the one power.

It looks like we cannot write it in the form Ax + By = C because the x has to be to the one power, not squared. So this is not a linear equation.

Graphing a Linear Equation
by Plotting Points

If the equation is linear:

Step 1: Find three ordered pair solutions.

You do this by plugging in ANY three values for x and find their corresponding y values.

Yes, it can be ANY three values you want, 1, -3, or even 10,000. Remember there are an infinite number of solutions. As long as you find the corresponding y value that goes with each x, you have a solution.

To review ordered pair solutions go to Tutorial 20: The Rectangular Coordinate System.

Step 2: Plot the points found in step 1.

Remember that each ordered pair corresponds to only one point on the graph.

The point lines up with both the x value of the ordered pair (x-axis) and the y value of the ordered pair (y-axis).

To review how to plot points on the graph go to Tutorial 20: The Rectangular Coordinate System.

Step 3: Draw the graph.

A linear equation will graph as a straight line.

If you know it is a linear equation and your points don’t line up, then you either need to check your math in step 1 and/or that you plotted all the points found correctly.

Example 3: Graph the linear equation y = 5x - 3.

Step 1: Find three ordered pair solutions.

I’m going to use a chart to organize my information. A chart keeps track of the x values that you are using and the corresponding y value found when you used a particular x value.

If you do this step the same each time, then it will make it easier for you to remember how to do it.

I usually pick out three points when I know I’m dealing with a line. The three x values I’m going to use are -1, 0, and 1. (Note that you can pick ANY three x values that you want. You do not have to use the values that I picked.) You want to keep it as simple as possible. The following is the chart I ended up with after plugging in the values I mentioned for x.

x	y = 5x - 3	(x, y)
-1	y = 5(-1) - 3 = -8	(-1, -8)
0	y = 5(0) - 3 = -3	(0, -3)
1	y = 5(1) - 3 = 2	(1, 2)

Step 2: Plot the points found in step 1.

Step 3: Draw the graph.

Example 4: Graph the linear equation

Step 1: Find three ordered pair solutions.

I’m going to use a chart to organize my information. A chart keeps track of the x values that you are using and the corresponding y value found when you used a particular x value.

If you do this step the same each time, then it will make it easier for you to remember how to do it.

x	y = 1/2x	(x, y)
-1	y = (1/2)(-1) = -1/2	(-1, -1/2)
0	y = (1/2)(0) = 0	(0, 0)
1	y = (1/2)(1) = 1/2	(1, 1/2)

Step 2: Plot the points found in step 1.

Step 3: Draw the graph.

Practice Problems

These are practice problems to help bring you to the next level. It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it. Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument. In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.

Practice Problems 1a - 1b: Determine whether the equation is linear or not.

1a. y = 2x - 1
(answer/discussion to 1a)

1b.

(answer/discussion to 1b)

Practice Problems 2a - 2b: Graph the linear equation.

2a. y = 2x - 1
(answer/discussion to 2a

2b.

(answer/discussion to 2b)

Need Extra Help on these Topics?

The following is a webpage that can assist you in the topics that were covered on this page:

http://www.purplemath.com/modules/graphlin.htm
This webpage helps you with graphing linear equations.

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

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