**Learning Objectives**

After completing this tutorial, you should be able to:

- Plot points on a rectangular coordinate system.
- Identify what quadrant or axis a point lies on.
- Know if an equation is a linear equation.
- Tell if an ordered pair is a solution of an equation in two variables or not.
- Graph an equation by plotting points.

** Introduction**

This section covers the basic ideas of graphing: **rectangular
coordinate system, solutions to equations in two variables, and
sketching
a graph.** Graphs are important in giving a visual
representation
of the correlation between two variables. Even though in this
section
we are going to look at it generically, using a general *x *and *y* variable, you can use two-dimensional graphs for any application where
you have two variables. For example, you may have a cost function
that is dependent on the quantity of items made. If you needed to
show your boss visually the correlation of the quantity with the cost,
you could do that on a two-dimensional graph. I believe that it
is
important for you learn how to do something in general, then when you
need
to apply it to something specific you have the knowledge to do
so.
Going from general to specific is a lot easier than specific to
general.
And that is what we are doing here looking at graphing in general so
later
you can apply it to something specific, if needed.

** Tutorial**

The following is the **rectangular coordinate system**:

It is made up of two number lines:
**origin** is where the two intersect. This is where both
number lines are 0.

- The horizontal number line is the
.*x*- axis - The vertical number line is the
.*y*- axis

It is split into four **quadrants** which are marked
on this graph
with Roman numerals.

Each point on the graph is associated with an **ordered
pair**.
When dealing with an *x, y* graph, *x *is
always first and *y* is always second in
the
ordered pair **( x, y)**. It is
a solution
to an equation in two variables. Even though there are two values
in the ordered pair, be careful that it associates to ONLY ONE point on
the graph,

Remember that each ordered
pair associates
with only one point on the graph. Just line up the *x* value and then the *y *value to get your
location.

**B(-1, 2) lies in quadrant II.**

**C(-3, -4) lies in quadrant III.**

**D(2, 0) lies on the x-axis.**

**E(0, 5) lies on the y-axis.**

The solutions to equations in two variables consist of
two values that
when substituted into their corresponding variables in the equation,
make
a true statement.

In other words, if your equation has two variables *x *and *y*,
and you plug in a value for *x* and its corresponding
value for *y* and the mathematical
statement
comes out to be true, then the *x *and *y* value that you plugged in would together be a solution to the
equation.

Equations in two variables can have more than one solution.

We usually write the solutions to equations in two
variables in ordered
pairs.

** Example
2**: Determine whether each ordered pair is a solution
of
the given equation. *y* = 5*x *- 7; (2, 3), (1, 5), (-1, -12)

Let’s start with the ordered pair (2, 3).

Which number is the *x* value and which one
is the *y *value? If you said *x* = 2 and *y *= 3, you are correct!

**Let’s plug (2, 3) into the equation and see what we
get:**

Now let’s take a look at (1, 5).

Which number is the *x* value and which one
is the *y* value? If you
said *x* = 1 and* y* = 5, you are right!

**Let’s plug (1, 5) into the equation and see what we
get:**

Whoops, it looks like we have ourselves **a FALSE
statement. This
means that (1, 5) is NOT a solution to the equation 5***x* - 7.

Now let’s look at (-1, -12).

Which number is the *x *value
and which one
is the *y *value? If
you said *x* = -1 and *y* = -12, you are right!

**Let’s plug (-1, -12) into the equation and see what
we get:**

We have another **TRUE statement. This means
(-1, -12) is another
solution to the equation ***y* = 5*x* - 7.

Note that you were only given three ordered pairs to check, however, there are an infinite number of solutions to this equation. It would very cumbersome to find them all.

Let’s start with the ordered pair (0, -3).

Which number is the *x* value and which one
is the *y *value? If you said *x* = 0 and *y *= -3, you are correct!

**Let’s plug (0, -3) into the equation and see what we
get:**

Now, let’s take a look at (1, -3).

Which number is the *x* value and which one
is the *y* value? If you
said *x* = 1 and* y* = -3, you are right!

**Let’s plug (1, -3) into the equation and see what we
get:**

Now, let’s look at (-1, -3).

Which number is the *x *value
and which one
is the *y *value? If
you said *x* = -1 and *y* = -3, you are right!

**Let’s plug (-1, -3) into the equation and see what we
get:**

**Standard Form:**

**A x + By = C**

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

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

This means we want to write it in the form A*x* + B*y* = C.

**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.

**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).

**Step 3: Draw the graph.**

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.

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

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

This means that we will have a line when we go to graph this.

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*.

**Step 1: Find six or seven
ordered pair solutions.**

Non-linear equations can vary on what the graph
looks like. So
it is good to have a lot of points so you can get the right shape of
the
graph.

**Step 2: Plot the points found
in step 1.**

**Step 3: Draw the graph.**

If the points line up draw a straight line
through them. If the
points are in a curve, draw a curve through them.

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 A*x* + B*y *= C because the *x* has to be to the one power, not squared. So **this is not a
linear
equation. **

However, we can still graph it.

The seven *x *values that
I'm going to use
are -3, -2, -1, 0, 1, 2, and 3. **(Note that you can pick ANY ***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*.

Note that the carrot top (^) represents an exponent. For
example, *x* squared can be written as *x*^2. The
second
column is showing you the 'scratch work' of how we got the
corresponding
value of *y*.

Do you think this equation is linear or not? It
is a tricky problem
because both the *x* and* y* variables are to the one power. **However, ***x* is inside the absolute value sign and we can’t just take it out of
there.

In other words, we can’t write it in the form A*x *+
B*y *= C. **This means that this
equation
is not a linear equation.**

If you are unsure that an equation is linear or not, you
can ALWAYS
plug in *x* values and find the
corresponding *y* values to come up with ordered pairs to plot.

The seven x values that I'm going to use are -3, -2,
-1, 0, 1, 2, and
3. **(Note that you can pick ANY ***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*.

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 Problem 1a:Plot the ordered pairs and name the quadrant or axis in which the point lies.

1a. A(3, 1), B(-2, -1/2), C(2, -2), and D(0,1)

(answer/discussion to 1a)

(answer/discussion to 1a)

Practice Problem 2a:Determine if each ordered pair is a solution of the given equation.

Practice Problems 3a - 3c:Determine whether each equation is linear or not. Then graph the equation.

** Need Extra Help on these Topics?**

**http://www.purplemath.com/modules/plane.htm**

This webpage helps you with plotting points.

**http://www.math.com/school/subject2/lessons/S2U4L1DP.html**

This website helps you with plotting points.

**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.**

Last revised on July 3, 2011 by Kim Seward.

All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.