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### Graphing y = mx + b

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Date: 01/20/97 at 10:58:13
From: Larry Shirley
Subject: x and y intercepts and graph

Dear Dr. Math,

I am having trouble understanding 3x + 2y = 5.  This has to be in
y = mx + b form.  After I get it in the proper form, how do I graph
it?

Sarah
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Date: 01/20/97 at 11:59:07
From: Doctor Lisa
Subject: Re: x and y intercepts and graph

Hi Sarah!

Let's take a look at your question step by step.  The first thing you
told me is that you need to get the equation into y = mx + b form.
So, to begin, you will work on solving this equation for y.  As you
may recall, what that means is that you want to manipulate the
equation so you get y by itself on one side of the equation.  This is
the process I would use:

3x + 2y = 5
2y = -3x + 5     (subtract 3x from each side to get 2y by itself)
y = -3/2x + 5/2 (divide both sides by 2 to get y by itself)

Now that I have the equation in the proper form, we're ready to
graph it.

Remember that y = mx + b is called "slope-intercept form."  If you
have an equation in this form (and written with the x-term first and
the constant second), you will have the slope and the y-intercept for
the graph.  The slope is "m" (the coefficient of x) and is the rise
over the run (or the change in y over the change in x).  The
y-intercept is "b" and is where the graph crosses the y-axis.

In this problem, m = -3/2 and b = 5/2.  I would first locate the
y-intercept on the graph.  Since b = 5/2, I would find where 5/2 (or
2 1/2) is on the y-axis and make a point.  This is where the graph
crosses the y-axis.

From there, I would use the slope to find other points on the graph.
When using slope, we first travel in the y direction and then move in
the x direction.  Keep in mind that a positive y direction is up, a
negative y direction is down, a positive x direction is right, and a
negative x direction is left.  When you have a positive slope, you
will either use a positive y and positive x movement or a negative y
and negative x direction. This is because a positive number divided by
a positive number is still positive and a negative number divided by a
negative number is also positive.  Since we have a negative slope, we
need one positive number and one negative number.

**  As I am describing the graphing process below, I am assuming
that you are using graph paper.  This process will work with any scale
on the graph (1 square could equal 2 units or 10 units, etc.).  **

Our slope is -3/2.  This means I can do one of two things: either go
up 3 squares (positive direction) and left 2 squares (negative
direction) or go down 3 squares (negative direction) and right 2
squares (positive direction).  Be careful with this problem because
you are beginning with a fraction (the 5/2 from above) - you'll have
to either estimate where halfway is or make each mark equal to 1/2 (in
other words, 2 marks on the graph = 1 unit).

Go to the 5/2 you marked earlier.  You can go up 3 squares from the
5/2 and then move over left 2 squares and make your next point.  You
can also go down 3 squares from the 5/2 and then move over right 2
squares and make your next point.  If you do both of those operations,
you will have 3 points and can connect the dots to make a line.  You
can count out as many points as you need to make the line, but I
suggest a minimum of 3 points for accuracy (although some teachers
want you to use 5 points).  The more points you have, the more
accurate the line will appear on your graph.

I hope I clearly answered your question for you and that this will
help you do other problems similar to it.  Have a great day!

-Doctor Lisa,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
Middle School Graphing Equations

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