Beginning Algebra
Tutorial 21:
Graphing Linear Equations
 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. See
graphing can be fun, it combines math and art together.
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
|
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
|
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
|
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. |
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.
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 = 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. |
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 are webpages
that can assist
you in the topics that were covered on this page:
This website helps you with graphing linear equations.
This webpage helps you with graphing linear equations.
|
for
some
more suggestions.
|
All contents
June 22, 2003 |
