**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.
- Tell if an ordered pair is a solution of an equation in two variables or not.
- Complete an ordered pair that has one missing value.

** Introduction**

This section covers the basic ideas of graphing: **rectangular
coordinate system, ordered pairs and solutions to equations in two
variables.**
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, the ** x coordinate
is** always first and the

A(2, 3), B(-1, 2), C(-3, -4), D(2, 0), and E(0, 5).

**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.**

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 ordered pair.

Since point A corresponds to 2 on the *x*-axis
and -3 on the *y*-axis, then **A’s
ordered pair
is (2, -3).**

Since point B corresponds to 3 on the *x*-axis
and 2 on the *y*-axis, then **B’s
ordered pair
is (3, 2).**

Since point C corresponds to -2 on the *x*-axis
and 3 on the *y*-axis, then **C’s
ordered pair
is (-2, 3).**

Since point D corresponds to -3 on the *x*-axis
and - 4 on the *y*-axis, then **D’s
ordered
pair is (-3, - 4)**.

Since point E corresponds to -3 on the *x*-axis
and 0 on the *y*-axis, then **E’s
ordered pair
is (-3, 0).**

Since point F corresponds to 0 on the* x*-axis
and 2 on the *y*-axis, then **F’s
ordered pair
is (0, 2).**

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
3**: 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.

x

This equation looks a little different than the one on
example 3.
In this equation, we only have an *x* value to
plug in. So as long as the* x* value
is
3, then we have a solution to the equation. It doesn’t matter
what *y*’s
value is.

Let’s start with the ordered pair (3, 5).

Which number is the *x* value and which one
is the *y *value? If you said *x* = 3 and *y *= 5, you are correct!

**Let’s plug (3, 5) into the equation and see what we
get:**

Now let’s take a look at (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 right!

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

Whoops, it looks like we have ourselves **a FALSE
statement. This
means that (2, 3) is NOT a solution to the equation ***x* = 3.

Now let’s look at (3, 4).

Which number is the *x *value
and which one
is the *y *value? If
you said *x* = 3 and *y* = 4, you are right!

**Let’s plug (3, 4) into the equation and see what we
get:**

We have another **TRUE statement. This means
(3, 4) is another
solution to the equation ***x* = 3.

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.

Given One Variable’s Value

Again, 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.

**Sometimes you are given a value of one of the
variables and you need
to find the corresponding value of the other variable. The steps
involved in doing that are:**

**Step 1: Plug given value for variable
into equation.**

**Step 2: Solve the equation for the
remaining variable.**

In the ordered pair (1, ), is 1 that is
given the* x* or the* y *value?

If you said *x*, you are
correct.

**Plugging in 1 for x into the given equation
and solving for y we get:**

***Solve for y**

In the ordered pair ( , -1), is the -1 that
is given the *x* or the* y* value?

If you said *y*, you are
correct.

**Plugging in -1 for y into the given equation
and solving for x we get:**

***Solve for x**

The only difference between this one and example 5
above is that we
are using a table to match up our values of our variables instead of
writing
it in an ordered pair. **The concept is still the same, we need
to find the corresponding values of our variables that are solutions to
the given equation.**

**Plugging in 0 for y into the given equation
and solving for x we get:**

**Plugging in -1 for y into the given equation
and solving for x we get:**

**Plugging in 1 for y into the given equation
and solving for x we get:**

**Filling in the table we get:**

** 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 Problem 1a:Plot each point 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:Find thex- andy- coordinates of the following labeled points.

2a.

Practice Problems 3a - 3b:Determine if each ordered pair is a solution of the given equation.

Practice Problem 4a:Complete each ordered pair so that it is a solution of the equation .

4a. (0, ) and
( , 1).

(answer/discussion to 4a)

(answer/discussion to 4a)

Practice Problem 5a:Complete the table of values for the equation .

5a.

** Need Extra Help on these Topics?**

**The following is a webpage
that can assist
you in the topics that were covered on this page: **

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

This webpage helps you with plotting points.

**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 29, 2011 by Kim Seward.

All contents copyright (C) 2001 - 2010, WTAMU and Kim Seward. All rights reserved.