Learning Objectives
Introduction
Tutorial
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).
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.
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 EquationsIn 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)
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:
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:
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.
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:
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:
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.
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.
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
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
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.
2a.
Practice Problems 3a - 3b:Determine if each ordered pair is a solution of the given equation.
3a. y = 4x - 10 ; (0, -10), (1, -14), (-1, -14)
(answer/discussion to 3a)
3b. y = -5 ; (2, -5), (-5, 1), (0, -5)
(answer/discussion to 3b)
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 .
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.
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