4.2: Graphs of Linear Equations
In Chapter 3, you learned how to solve equations in one variable. The answer was of the form variable = some number. In this lesson, you will learn how to solve equations with two variables. Below are several examples of two-variable equations:
\begin{align*}p& =20(h)\\ m& =8.15(h)\\ y& =4x+7\end{align*}
You have seen each of these equations in a previous lesson. 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 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 \begin{align*}(y)\end{align*} to hire a taxi and the distance traveled in miles \begin{align*}(x)\end{align*}: \begin{align*}y=0.8x+3\end{align*}.
This is an equation in two variables. By creating a table, we can graph these ordered pairs to find the solutions.
\begin{align*}x\end{align*} (miles) | \begin{align*}y\end{align*} (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 \begin{align*}y-\end{align*}ordinate of the desired \begin{align*}x-\end{align*}abscissa.
Multimedia Link: To see more simple examples of graphing linear equations by hand, view the video
Khan Academy graphing lines (9:49)
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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.
Graphs of Horizontal and Vertical Lines
Not all graphs are slanted or oblique. Some are horizontal or vertical. Read through the next situation to see why.
Example: “Mad-cabs” have an unusual offer going on. They are charging $7.50 for a taxi ride of any length within the city limits. Graph the function that relates the cost of hiring the taxi \begin{align*}(y)\end{align*} to the length of the journey in miles \begin{align*}(x)\end{align*}.
Solution: No matter the mileage, your cab fare will be $7.50. To see this visually, create a graph. You can also create a table to visualize the situation.
# of miles \begin{align*}(x)\end{align*} | Cost \begin{align*}(y)\end{align*} |
---|---|
0 | 7.50 |
10 | 7.50 |
15 | 7.50 |
25 | 7.50 |
60 | 7.50 |
Because the mileage can be anything, the equation should relate only to the restricted value, in this case, \begin{align*}y\end{align*}. The equation that represents this situation is:
\begin{align*}y=7.50\end{align*}
Whenever there is an equation of the form \begin{align*}y= \text{constant}\end{align*}, the graph is a horizontal line that intercepts the \begin{align*}y-\end{align*}axis at the value of the constant.
Similarly, if there is an equation of the form \begin{align*}x=\text{constant}\end{align*}, the graph is a vertical line that intercepts the \begin{align*}x-\end{align*}axis at the value of the constant. Notice that this is a relation but not a function because each \begin{align*}x\end{align*} value (there’s only one in this case) corresponds to many (actually an infinite number) \begin{align*}y\end{align*} values.
Example 1: Plot the following graphs.
(a) \begin{align*}y=4\end{align*}
(b) \begin{align*}y=-4\end{align*}
(c) \begin{align*}x=4\end{align*}
(d) \begin{align*}x=-4\end{align*}
Solution:
(a) \begin{align*}y=4\end{align*} is a horizontal line that crosses the \begin{align*}y-\end{align*}axis at 4.
(b) \begin{align*}y=-4\end{align*} is a horizontal line that crosses the \begin{align*}y-\end{align*}axis at –4.
(c) \begin{align*}x=4\end{align*} is a vertical line that crosses the \begin{align*}x-\end{align*}axis at 4.
(d) \begin{align*}x=-4\end{align*} is a vertical line that crosses the \begin{align*}x-\end{align*}axis at –4.
Analyzing Linear Graphs
Analyzing linear graphs is a part of life – whether you are trying to decide to buy stock, figure out if your blog readership is increasing, or predict the temperature from a weather report. Although linear graphs can be quite complex, such as a six-month stock graph, many are very basic to analyze.
The graph below shows the solutions to the price before tax and the price after tax at a particular store. Determine the price after tax of a $6.00 item.
By finding the appropriate \begin{align*}x-\end{align*}abscissa ($6.00), you can find the solution, the \begin{align*}y-\end{align*}ordinate (approximately $6.80). Therefore, the price after tax of a $6.00 item is approximately $6.80.
The following graph shows the linear relationship between Celsius and Fahrenheit temperatures. Using the graph, convert \begin{align*}70^\circ F\end{align*} to Celsius.
By finding the temperature of \begin{align*}70^\circ F\end{align*} and locating its appropriate Celsius value, you can determine that \begin{align*}70^\circ F \approx 22^\circ C\end{align*}.
Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Graphs of Linear Equations (13:09)
- What are the solutions to an equation in two variables? How is this different from an equation in one variable?
- What is the equation for the \begin{align*}x-\end{align*}axis?
- What is the equation for the \begin{align*}y-\end{align*}axis?
- Using the tax graph from the lesson, determine the net cost of an item costing $8.00 including tax.
- Using the temperature graph from the lesson, determine the following:
- The Fahrenheit temperature of \begin{align*}0^\circ C\end{align*}
- The Fahrenheit temperature of \begin{align*}30^\circ C\end{align*}
- The Celsius temperature of \begin{align*}0^\circ F\end{align*}
- The Celsius equivalent to water boiling \begin{align*}(212^\circ F\end{align*})
- Graph the following equations on separate Cartesian planes.
- \begin{align*}y=-2\end{align*}
- \begin{align*}7=x\end{align*}
- \begin{align*}4.5=y\end{align*}
- \begin{align*}x=8\end{align*}
The graph below shows a conversion chart for converting between the weight in kilograms to weight in pounds. Use it to convert the following measurements.
- 4 kilograms into weight in pounds
- 9 kilograms into weight in pounds
- 12 pounds into weight in kilograms
- 17 pounds into weight in kilograms
Write the equations for the graphed lines pictured below.
- \begin{align*}E\end{align*}
- \begin{align*}B\end{align*}
- \begin{align*}C\end{align*}
- \begin{align*}A\end{align*}
- \begin{align*}D\end{align*}
- At the airport, you can change your money from dollars into Euros. The service costs $5, and for every additional dollar you get 0.7 Euros. Make a table for this information and plot the function on a graph. Use your graph to determine how many Euros you would get if you give the exchange office $50.
- Think of a number. Triple it, and then subtract seven from your answer. Make a table of values and plot the function that this sentence represents.
Find the solutions to each equation by making a table and graphing the coordinates.
- \begin{align*}y=2x+7\end{align*}
- \begin{align*}y=0.7x-4\end{align*}
- \begin{align*}y=6-1.25x\end{align*}
Mixed Review
- Find the percent of change: An item costing $17 now costs $19.50.
- Give an example of an ordered pair located in Quadrant III.
- Jodi has \begin{align*}\frac{1}{3}\end{align*} of a pie. Her little brother asks for half of her slice. How much pie does Jodi have?
- Solve for \begin{align*}b: b+16=3b-2\end{align*}.
- What is 16% of 97?
- Cheyenne earned a 73% on an 80-question exam. How many questions did she answer correctly?
- List four math verbs.