4.2: Graphs of Linear Equations
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.
Watch This
To see some 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.
Guidance
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:
\begin{align*}p& =20(h)\\ m& =8.15(n)\\ y& =4x+7\end{align*}
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 \begin{align*}p=2(h)\end{align*}.
Solution:
Make a table and then graph the points:
\begin{align*}h\end{align*} | \begin{align*}p\end{align*} |
---|---|
0 | 0 |
1 | 2 |
2 | 4 |
3 | 6 |
Example B
Graph \begin{align*} y=4x+7\end{align*}.
Solution:
Make a table and then graph the points.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
-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 \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*} of the desired \begin{align*}x\end{align*}.
Guided Practice
Graph \begin{align*}m=8.15(n)\end{align*}.
Solution:
To graph this linear equation, we will make a table of some points. We can plug in values for \begin{align*}h\end{align*} to get values for \begin{align*}m\end{align*}. For example:
\begin{align*}m=8.15(-1)=-8.15\end{align*}
\begin{align*}n\end{align*} | \begin{align*}m\end{align*} |
---|---|
\begin{align*}-1\end{align*} | \begin{align*}-8.15\end{align*} |
\begin{align*}0\end{align*} | \begin{align*}0\end{align*} |
\begin{align*}1\end{align*} | \begin{align*}8.15\end{align*} |
\begin{align*}2\end{align*} | \begin{align*}16.3\end{align*} |
Next, we will graph each point, and then connect the points with a line.
Practice
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?
- 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.
Graph the solutions to each linear 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 sum: \begin{align*}\frac{3}{8}+\frac{1}{5}-\frac{5}{9}\end{align*}.
- Solve for \begin{align*}m: 0.05m+0.025(6000-m)=512\end{align*}.
- Solve the proportion for \begin{align*}u: \frac{16}{u-8}=\frac{36}{u}\end{align*}.
- What does the Additive Identity Property allow you to do when solving an equation?
- Shari has 28 apples. Jordan takes \begin{align*}\frac{1}{4}\end{align*} of the apples. Shari then gives away 3 apples. How many apples does Shari have?
- The perimeter of a triangle is given by the formula \begin{align*}Perimeter=a+b+c\end{align*}, where \begin{align*}a, b,\end{align*} and \begin{align*}c\end{align*} are the lengths of the sides of a triangle. The perimeter of \begin{align*}\triangle ABC\end{align*} is 34 inches. One side of the triangle is 12 inches. A second side is 7 inches. How long is the remaining side of the triangle?
- Evaluate \begin{align*}\frac{y^2-16+10y+2x}{2}\end{align*}, for \begin{align*}x=2\end{align*} and \begin{align*}y=-2.\end{align*}
linear equation
A linear equation has solutions that are the coordinates on a graphed line.Cartesian Plane
The Cartesian plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin.Slope
Slope is a measure of the steepness of a line. A line can have positive, negative, zero (horizontal), or undefined (vertical) slope. The slope of a line can be found by calculating “rise over run” or “the change in the over the change in the .” The symbol for slope isImage Attributions
Here you'll learn how to use coordinate pairs to graph linear equations.