Beginning Algebra
Tutorial 20: The Rectangular Coordinate System
 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.
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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
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Rectangular Coordinate System
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The following is the rectangular coordinate system: |
It is made up of two number lines:
- The horizontal number line is the x-
axis.
- The vertical number line is the y-
axis.
The origin is where the two intersect. This is where both
number lines are 0. 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 y coordinate 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, 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). |
 Example
1: Plot the ordered pairs and name the
quadrant
or axis in which the point lies.
A(2, 3), B(-1,
2), C(-3, -4), D(2, 0), and E(0, 5). |
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. |
A(2, 3) lies in quadrant I.
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. |
Example
2: Find the x-
and y- coordinates of the following
labeled
points
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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). |
Solutions of Equations
in Two Variables
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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
3: Determine whether each ordered pair is a solution
of
the given equation.
y =
5x - 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: |
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*Plug in 2 for x and
3 for y
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This is a TRUE statement, so (2, 3) is a solution to
the equation y = 5x - 7.
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: |
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*Plug in 1 for x and 5 for y
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Whoops, it looks like we have ourselves a FALSE
statement. This
means that (1, 5) is NOT a solution to the equation 5x - 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: |
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*Plug in -1 for x and -12 for y
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We have another TRUE statement. This means
(-1, -12) is another
solution to the equation y = 5x - 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. |
Example
4: Determine whether each ordered pair is a solution
of
the given equation.
x = 3; (3, 5), (2, 3), (3, 4) |
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: |
This is a TRUE statement, so (3, 5) is a solution to
the equation x = 3.
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. |
Finding the Corresponding Value
in an Ordered
Pair
Given One Variable’s Value
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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. |
Example
5: Complete each ordered pair so that it is a solution
of the equation  .
(1, ) and ( , -1). |
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: |
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*Plug in 1 for x
*Solve for y
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So, the ordered pair (1, 1) would be a
solution to the given
equation.
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: |
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*Plug in -1 for y
*Solve for x
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So, the ordered pair (4, -1) would be another
solution to the given
equation. |
Example
6: Complete the table of values for the equation  .
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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: |
So, the ordered pair (-1/2, 0) would be a solution
to the given
equation.
Plugging in -1 for y into the given equation
and solving for x we get: |
So, the ordered pair (-1/2, -1) would be another
solution to the
given equation.
Plugging in 1 for y into the given equation
and solving for x we get: |
So, the ordered pair (-1/2, 1) would be another
solution to the
given equation.
Filling in the table we get:
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x
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y
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-1/2
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0
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-1/2
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-1
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-1/2
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1
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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.
Practice
Problem 2a: Find the x- and y-
coordinates of the
following labeled points.
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  .
Practice
Problem 5a: Complete the table of values for the
equation  .
Need Extra Help on these Topics?
Last revised on July 29, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2010, WTAMU and Kim Seward. All rights reserved.
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