Ordered Pairs
Often, to get an idea of the behavior of an equation, we will make a picture
that represents the solutions to the equation. A graph gives us that picture.
The rectangular coordinate plane, where we place our graph, is created by
a horizontal number line (x-axis) and a vertical number line (y-axis).
Where the two number lines meet in the center, at x=0 and y=0. is called
the origin. Notice that the rectangular coordinate plane has four sections
which are called quadrants.
y
6
4
II
I
2
6
4
2
III
2
2
4
6
x
IV
4
6
A point is an ordered pair given as (x, y). The first number is the value
on the x-axis. This is the distance the point moves right (if positive) or
left (if negative) from the origin. The second number is the value on the
y-axis. This is the distance the point moves up (if positive) or down (if
negative) from the origin.
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 1: Give the coordinates of the points on the graph.
y
6
B
4
A
6
4
2
2
2
2
C
4
6
x
D
4
6
E
Point A: From the point down to the x-axis = -4
From the point right to the y-axis = 2
The coordinates are (-4, 2)
Point B: From the point down to the x-axis = 4
From the point left to the y-axis = 5
The coordinates are (-4, 5)
Point C: The point is on the x-axis = 3
The point left to the y-axis = 0
The coordinates are (3, 0)
Point D: From the point up to the x-axis = 1
From the point left to the y-axis = -3
The coordinates are (1, -3)
Point E: From the point up to the x-axis = 0
The point is on the y-axis = -5
The coordinates are (0, -5)
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
In a similar manner, we can go backwards and plot points on the plane.
Example 2: Plot the points:
A=(5, 4), B=(-2, -5), C=(-4, 5), D=(0, 2), E=(3, 0)
y
6
C
A
4
D2
6
4
2
2
E
4
6
x
2
B
4
6
A is at (5, 4), so x=5 (right 5) and y=4 (up 4)
B is at (-2, -5), so x=-2 (left 2) and y=-5 (down 5)
C is at (-4, 5), so x=-4 (left 4) and y=5 (up 5)
D is at (0, 2), so x=0 (no movement) and y=2 (up 2)
E is at (3, 0), so x=3 (right 3) and y=0 (no movement)
The main purpose of graphs is not to plot random points, but rather to give
a picture of the solutions to an equation. Each point on the graph of a line
is solution to that linear equation.
To check whether a point is a solution to the linear equation, substitute
the x-value of the point into the equation in place of x, and substitute the
y-value of the point into the equation in place of y. Then simplify. A true
final statement indicates that the point is a solution. A false final
statement indicates that the point is not a solution.
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 3: Are the points (-1, 7) and (-2, 3) solutions to the equation
y = -2x + 5
For (-1, 7): 7 = -2(-1) + 5
7=2+5
7=7
True, (-1, 7) is a solution
For (-2, 3): 3 = -2(-2) + 5
3=4+5
3≠9
False, (-2, 3) is not a solution
1
Example 4: Are the point (-1, 1) and (3, ) solutions to the equation
5
3x – 5y = 8
For (1, 1): 3(1) – 5(1) = 8
-3 – 5 = 8
-8 ≠ 8
False, (-1, 1) is not a solution
1
1
For (3, ): 3(3) – 5( ) = 8
5
5
9–1=8
8=8
1
True, (3, ) is a solution
5
To find the points of a linear equation, we need to construct ordered pairs.
If we are given an x- value, we substitute that x-value into the linear
equation and solve for y. This gives the y-value and completes the ordered
pairs. In a similar manner, if we are given the y-value, we substitute to
find the x- value and complete the ordered pair.
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 5: Complete the following ordered pairs for 2x – 3y = 6
(-3, ___), (0, ___), (___, 1), (3, ___)
2(-3) – 3y = 6
- 6 – 3y = 6
+6
+6
Add 6 to both sides
- 3y = 12
-3
-3
Divide both sides by -3
y = -4
The ordered pair is (-3, -4)
2(0) – 3y = 6
0 - 3y = 6
- 3y = 6
-3 -3
Divide both sides by -3
y = -2
The ordered pair is (0, -2)
2x - 3(1) = 6
2x - 3 = 6
+3 +3
2x
=9
2
2
x
=
9
2
Add 3 to both sides
Divide both sides by 2
9
The ordered pair is ( , 1)
2
2(3) – 3y = 6
6 – 3y = 6
-6
-6
Subtract 6 from both sides
- 3y = 0
-3 -3
Divide both sides by -3
y=0
The ordered pair is (3, 0)
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 6: Complete the ordered pairs for y = 2x – 1
(0, ___), (-4, ___), (___, 1), (___, 0)
y = 2(0) – 1
y=0–1
y=-1
The ordered pair is (0, -1)
y = 2(-4) – 1
y=-8–1
y=-9
The ordered pair is (-4, -9)
1 = 2x – 1
+1
+1
Add 1 to both sides
2 = 2x
2 2
Divide both sides by 2
1=x
The ordered pair is (1, 1)
0 = 2x – 1
+1
+1
1 = 2x
2 2
1
2
Add 1 to both sides
Divide both sides by 2
=x
1
The ordered pair is ( , 0)
2
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)