7.8 Graphs of Linear Equations
Learning Objectives
- I can use coordinate pairs to graph linear equations.
Introduction
Suppose that there is a linear relationship between your annual income and the amount you must pay in state income tax. What if you wanted to make a graph that showed how much tax you must pay based on your income? Could you do it? After completing this Concept, you'll be able to take a linear equation like the one representing the situation just described and graph it on a coordinate plane.
Guided Learning
To see some simple examples of graphing linear equations by hand, view the video
Khan Academy graphing lines video (9:49).
The narrator of the video models graphing linear equations by using a table of values to plot points and then connecting those points with a line. This process reinforces the procedure of graphing lines by hand.
Previously, you learned how to solve equations in one variable. The answer was of the form variable = some number. In this Concept, you will learn how to solve equations with two variables. Below are several examples of two-variable equations:
You have seen each of these equations used in a previous Concept, and you will see how to graph them in this Concept. Their solutions are not one value because there are two variables. The solutions to these equations are pairs of numbers. These pairs of numbers can be graphed in a Cartesian plane.
The solutions to an equation in two variables are sets of ordered pairs.
The solutions to a linear equation are the coordinates on the graphed line.
By making a table, you are finding the solutions to the equation with two variables.
Example A
Graph .
Solution:
Make a table and then graph the points:
h | p |
0 | 0 |
1 | 2 |
2 | 4 |
3 | 6 |
Example B
Graph .
Solution:
Make a table and then graph the points.
x | y |
-1 | 3 |
0 | 7 |
1 | 13 |
Example C
A taxi fare costs more the further you travel. Taxis usually charge a fee on top of the per-mile charge. In this case, the taxi charges $3 as a set fee and $0.80 per mile traveled. Find all the possible solutions to this equation.
Solution: Here is the equation linking the cost in dollars to hire a taxi and the distance traveled in miles : .
This is an equation in two variables. By creating a table, we can graph these ordered pairs to find the solutions.
x (miles) | y (cost $) |
0 | 3 |
10 | 11 |
20 | 19 |
30 | 27 |
40 | 35 |
The solutions to the taxi problem are located on the green line graphed above. To find any cab ride cost, you just need to find the of the desired .
Guided Practice
Graph . Check your work with a partner.
Solution:
To graph this linear equation, we will make a table of some points. We can plug in values for to get values for . For example:
n | m |
-1 | -8.15 |
0 | 0 |
1 | 8.15 |
2 | 16.3 |
Next, we will graph each point, and then connect the points with a line.
Review
- To graph linear equations, find solutions for the 2 values, list them in a table, then graph the values.
Linear Equation
A linear equation has solutions that are the coordinates on a graphed line.
Additional Resources
Graphs of Linear Equations Video