Learning Objectives
Introduction
In this tutorial we will dive even deeper into linear equations. We will be going over how to come up with our own equations given certain information. After you finish tutorials 12 - 17, you will be an old pro at linear equations and graphing. Let's see what we can do with slopes, y-intercepts and linear equations.
Tutorial
OR
Example 1: Use the slope/intercept form of the linear equation to write the equation of a line with slope - 4; y -intercept (0, ½).
Nothing to it.
Recall in Tutorial 14: Graphing Linear Equations that the x value on the y-intercept is always 0.
Positive Slope
If your slope is positive, then you are going to go in
the same
direction for both the rise and the run. In other words, you
either go in a positive direction for rise (up) and a positive
direction
for run (right) OR a negative direction for rise (down) and a negative
direction for run (left).
For example, if the slope is 2/3, then you can rise up 2
and run right
3 from any point that is on the line. Or, you can go down 2 and
run
left 3 from any point that is on the line.
Negative Slope
If your slope is negative, then you are going to go in
opposite
directions for the rise and the run. In other words, you
either
go in a negative direction for rise (down) and a positive direction for
run (right) OR a positive direction for rise (up) and a negative
direction
for run (left).
For example, if the slope is -2/3, then you can go down
2 and run right
3 from any point that is on the line. Or, you can rise 2 and run
left 3 from any point that is on the line.
Integer Slopes
If your slope is an integer,
remember that the denominator is understood to be 1. So, the run
part of an integer slope is going to be 1. For example, if the
slope
is 3, you want to think of it as 3/1. You would rise up 3 and run
right 1 from any point on the line to get another point on the line.
Let’s look at some examples for this new method.
*Inverse of mult. by -2 is div.
by -2
*Slope/intercept form of the line
I got m = 3/2 and y-intercept = -2.
This linear equation is already in the slope/intercept form.
Note how we are missing a constant being added to the x term. If we are missing that constant, what is it understood to be???
The slope is -1 and the y-intercept is 0.
A line going through
the point and
having slope of m
would have the equation
We can use this form when we need to come up with a linear equation and we don’t know the y-intercept.
No matter what form that you end up using, keep in mind that you ALWAYS need two pieces of information when you go to write an equation:
Looks like we have all the information we need. We are ready to put our equation together. Since we don’t have the y-intercept, we will have to use the point/slope form since that is set up for ANY point (not just the y-int.).
*Slope/intercept form of the line
We have more than enough points. However, what about the slope? Does this mean we can’t work out the problem? You are not going to get off that easily. We do have a way of finding the slope. Tutorial 15: The Slope of a Line shows us how we can get the slope given two points.
Let’s find that slope:
*Plug in values
*Simplify
*Inverse of sub. 3 is add 3
*LCD is 4
*Mult. 3/1 by 4/4 to get
12/4
*Slope/intercept form of the line
*Function notation
We have our point. However, what about the slope?
We need to do a little digging to get it.
Recall that Tutorial
15: The Slope of a Line tells us that parallel lines have the same slope.
So, if we know the slope of the line parallel to our line, we have it
made.
Find the slope of the parallel line:
OK, now we have our slope, which is -3. Now it is just like examples 4 and 5 above, we want to put the slope and one point into the point/slope equation.
*Slope/intercept form of the
line
*Function notation
We have our point. However, what about the slope?
We need to do a little work in that department.
Recall that Tutorial
15: The Slope of a Line tells us that the slopes of perpendicular lines
are negative reciprocals of each other. So, if we
know
the slope of the line perpendicular to our line, we have it made.
Find the slope of the perpendicular line:
*Inverse of mult. by -2 is div. by -2
*Slope/intercept form of the line
What did you come up with?
I came up with -2 for the slope of our line.
Now we can go on to the equation of our line:
*Slope/intercept form of the
line
*Function notation
Since it passes through (4, -1), and a horizontal line is in the form y = c, where the y value is ALWAYS equal to the same value throughout, this means our equation would have to be y = -1.
Note that -1 is the y value of the ordered pair given.
Since it passes through (2, 3), and a vertical line is in the form x = c, where the x value is ALWAYS equal to the same value throughout, this means our equation would have to be x = 2.
Note that 2 is the x value of the ordered pair given.
Practice Problems
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 Problem 1a: Use the slope-intercept form of the linear equation to write the equation of the line.
Practice Problem 2a: Graph the linear equation using the point/slope form.
2a. 3x + 5y = 10
(answer/discussion
to 2a)
Practice Problem 3a: Find the equation of the line with the given slope and containing the given point. Write the equation in slope-intercept form.
Practice Problem 4: Find the equation of a line passing through the given points. Use function notation to write the equation.
4a. (0, 0) and (5, 10)
(answer/discussion
to 4a)
Practice Problems 5a - 5b: Write an equation of the line.
Practice Problems 6a - 6b: Find an equation of the line. Write the equation using function notation.
6a.Passes through (2, 3) and
parallel to 5x + 2y = 4
(answer/discussion
to 6a)
Need Extra Help on these Topics?
http://www.math.com/school/subject2/lessons/S2U4L3DP.html
This website helps you with graphing linear equations.
http://www.purplemath.com/modules/slopgrph.htm
This website covers graphing linear equations using slopes.
http://www.purplemath.com/modules/slopyint.htm
This website goes over the meaning of slope and y-intercept.
http://www.purplemath.com/modules/strtlneq.htm
This website helps you with writing linear equations.
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
Last revised on July 5, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.