7th grade math How To Graph Linear Equations

HOW TO GRAPH HOW TO
LINEAR AND
INEQUALITY The document shows the 4 steps to graphing a linear
EQUATIONS equation as well as the additional steps to graph an
inequality.
[Document subtitle]
P Denise Smith of TutorByDenise
404-939-3806

www.tutorbydenise.com

HOW TO GRAPH LINEAR AND INEQUALITY EQUATIONS

Graphing Linear Equations:
Making a T-chart (finding plot points) (page 1 of 4)

Sections: Making a T-chart, Plotting the points and drawing the line, Examples

Graphing linear equations is pretty simple, but only if you work neatly. If you're messy, you'll often make
extra work for yourself, and you'll frequently get the wrong answer. I'll walk you through a few examples.
Follow my pattern, and you should do fine.

 Graph y = 2x + 3 (so 2x +3 is your function and x and y are the ordered pairs)

First, you draw what is called a "T-chart": it's a
chart that looks a bit like the letter "T":

The left column will contain the x-values that

you will pick, and the right column will contain

the corresponding y-values that you will

compute.

Label the columns:

The first column will be where you choose your

input (x) values; the second column is where
you find the resulting output (y) values.
Together, these make a point, (x, y).

Pick some values for x. It's best to pick at least three value, to verify (when you're graphing) that
you're getting a straight line. ("Linear" equations, the ones with just an x and a y, with no squared

variables or square-rooted variables or any other fancy stuff, always graph as straight lines.
That's where the name "linear" came from!)

Which x-values you pick is totally up to you! And it's perfectly okay if you pick values that are

different from the book's choices, or different from your study partner's choices, or different from
my choices. Some values may be more useful than others, but the choice is entirely up to you.

Then your y-values will come from evaluating the equation at the x-values you've chosen. And

the T-chart keeps the information all nice and neat.

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HOW TO GRAPH LINEAR AND INEQUALITY EQUATIONS

I'll pick the following x-values:

Copyright © Elizabeth Stapel 2000-2011
All Rights Reserved

You can pick whatever values you like, but it's often best to "space them out" a bit. For instance,

picking x = 1, 2, 3 might not give you as good a picture of your line as picking x = –3, 0, 3.

That's not a rule, but it's often a helpful method.

Once you've picked x-values, you

have to compute the

corresponding y-values:

Some people like to add a third column to their T-chart to give room for a clear listing of the points that
they've found:

Which format you use is (usually) just a matter of taste. Unless your instructor specifies, either format
should be fine.

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HOW TO GRAPH LINEAR AND INEQUALITY EQUATIONS

Graphing Linear Equations: Plotting Your Points

Now that you have your points, you need to draw your axes. REMEMBER TO USE YOUR RULER! If you
don't use a ruler, you will have messy axes and inconsistent scales on the axes, and your points will NOT
line up properly. Don't "fake it" with your graphs. Get in the habit now of drawing neatly. It will save you so
much trouble down the line! (And, no, using graph paper is not the same as, nor does it replace, using a
ruler!)

Also, make sure you draw your axes large enough that your graph will be easily visible. On a standard-
sized sheet of paper (8.5 by 11 inches, or A4), you will be able to fit two or three graphs on a page. If you
are fitting more than three graphs on one side of a sheet, then you're probably drawing them too small.

Here are my
axes:

Remember that the arrows indicate the direction in which the values are increasing. Your book (and even
your teacher) may draw things incorrectly, but that's no excuse for you. Arrows go on the upper numerical
ends of each axis, and NOWHERE else.
Once I've drawn my axes, I have to label them with an appropriate scale. "Appropriate" means "one that
is neat and that fits the numbers I'm working with". For instance, considering the values I'm working with,
I'll count off by ones. And ALWAYS use a ruler to make sure that your tick-marks are even!

Here's my
scale:

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HOW TO GRAPH LINEAR AND INEQUALITY EQUATIONS

Graphing Linear Equations: Plotting Your Points and Drawing Your Line
Note that I've made every fifth tick-mark a bit longer. This isn't a rule, but I've often found it helpful for
counting off the larger points; it's more of a time-saver than anything else.

By the way, if you don't use a ruler or take a
modicum of care in your work, your graphs will
look like what math instructors are more
accustomed to seeing:

I'm not kidding; people really hand in "work" that looks like this. Please don't be one of those people!

Now I'll plot (draw) the points I'd computed in
my T-chart:

...and then I'll finish up by connecting the
dots:

Since "linear" equations graph as straight, you might as well use your ruler for this part, too. The drawing

.is the answer they're looking for. Once you've connected your dots, you're done with the exercise

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HOW TO GRAPH LINEAR AND INEQUALITY EQUATIONS

Graphing Linear Equations: Examples

Sections: Making a T-chart, Plotting the points and drawing the line, Examples

 Graph y = (–5/3)x – 2

First I'll do the T-chart.

Since I am multiplying x by a fraction, I will
pick x-values that are multiples of 3, so the

denominator will cancel out and I won't have
fractions.

Then I'll plot my points and draw my graph:

 Graph y = 7 – 5x

Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

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HOW TO GRAPH LINEAR AND INEQUALITY EQUATIONS

First I'll do the T-chart.

This equation is an example of a situation in which you will probably want to be particular about

the x-values you pick. Because the x is multiplied by a relatively large value, the y-values grow
quickly. For instance, you probably wouldn't want to use x = 5 or x = –3. You could pick larger
x-values if you wished, but your graph would get awfully tall.

And as you can see, the
graph is pretty tall already:

 Graph y = 3
Don't let this one scare you. Yes, there is no "x" in the equation, but that's okay. Just think about
it this way: it doesn't matter what x-value you pick; y will always be 3.

Your T-chart would look something like this:

Then your graph would look like this: 404-939-3806 6

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HOW TO GRAPH LINEAR AND INEQUALITY EQUATIONS

Any time you have a "y equals a number" equation, with no x in it, the graph will always be a horizontal

line.

Graphing Linear Inequalities: y > mx + b, etc

You begin by using the same steps you used to graph a linear equation
 Make your t-chart (table)
 Insert x value
 Solve
 Plot points

BUT there is a difference when you draw your line. If the sign is
 < or > you draw a dotted line (the bar under the symbol means equal to)
 < or > you draw a solid line

Now because it says greater than or less than we have to shade an area either above or below the line
How do we determine that

The steps for graphing two-variable linear inequalities are very much the same.
 Graph the solution to y < 2x + 3.

Just as for number-line inequalities, my first step is to find the "equals" part. For two-variable linear inequalities,
the "equals" part is the graph of the straight line; in this case, that means the "equals" part is the line y = 2x + 3:

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HOW TO GRAPH LINEAR AND INEQUALITY EQUATIONS

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Now we're at the point where your book probably gets
complicated, with talk of "test points" and such. Just
pick an ordered pair, any ordered pair and replace the
x and y in the equation with the order pair you chose. If your answer makes y: y < 2x + 3 a true equation shade in
everything on that side, if it is not correct shade the other side.

I've already graphed the "or equal to" part (it's just the line); now I'm ready to do the "y less than" part. In other
words, this is where I need to shade one side of the line or the other. Now think about it: If I need y LESS THAN the
line, do I want ABOVE the line, or BELOW? Naturally, I want below the line. So I shade it in:

And that's all there is to it: the side I shaded is the "solution region" they want.

This technique worked because we had y alone on one side of the inequality. Just as with plain old lines, you
always want to "solve" the inequality for y on one side.

 Graph the solution to 2x– 3y < 6.

First, I'll solve for y:

2x – 3y < 6
–3y < –2x + 6
y > ( 2/3 )x – 2

[Note the flipped inequality sign in the last line. I mustn't forget to flip the inequality if I multiply or divide through
by a negative!]

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HOW TO GRAPH LINEAR AND INEQUALITY EQUATIONS

Now I need to do the same steps as before for the line y = ( 2/3 )x – 2. It looks like this:

But this exercise is a inequality that isn't an "or equals to" inequality When I have an inequalities on the number
line that does not include “or equal to”, I draw a dashed line. So the border of my solution region actually looks
like this:

By using a dashed line, I still know where the border is, but I also know that the border isn't included in the
solution. Since this is a "y greater than" inequality, I want to shade above the line, so my solution looks like this:

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