4.3: Graphs Using Intercepts
As you may have seen in the previous lesson, graphing solutions to an equation of two variables can be timeconsuming. Fortunately, there are several ways to make graphing solutions easier. This lesson will focus on graphing a line by finding its intercepts. Lesson 4.5 will show you how to graph a line using its slope and
In geometry, there is a postulate that states, “Two points determine a line.” Therefore, to draw any line, all you need is two points. One way is to find its intercepts.
An intercept is the point at which a graphed equation crosses an axis.
The
The
By finding the intercepts of an equation, you can quickly graph all the possible solutions to the equation.
Finding Intercepts Using Substitution
Remember that the Substitution Property allows the replacement of a variable with a numerical value or another expression. You can use this property to help find the intercepts of an equation.
Example: Graph
Solution: The
Continue solving for
The
Repeat the process to find the
The
To graph the line formed by the solutions of the equation
Example: Graph
Solution: Determine the
The ordered pair of the
Finding Intercepts Using the CoverUp Method
By finding an intercept, you are substituting the value of zero in for one of the variables.
To find the
To find the
A second method of finding the intercepts is called the CoverUp Method. Using the Multiplication Property of Zero
Example: Graph
Solution: To solve for the
To solve for the
Now graph by first plotting the intercepts then drawing a line through these points.
Example 1: Jose has $30 to spend on food for a class barbeque. Hot dogs cost $0.75 each (including the bun) and burgers cost $1.25 (including bun and salad). Plot a graph that shows all the combinations of hot dogs and burgers he could buy for the barbecue, spending exactly $30.
Solution: Begin by translating this sentence into an algebraic equation. Let
Find the intercepts of the graph. This example will use the CoverUp Method. Feel free to use Substitution if you prefer.
By graphing Jose’s situation, you can determine the combinations of hot dogs and burgers he can purchase for exactly $30.00.
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. CK12 Basic Algebra: Graphing Using Intercepts (12:18)
 Define intercept.
 What is the ordered pair for an
x− intercept?  Explain the process of the CoverUp Method.
Find the intercepts for the following equations using substitution.

y=3x−6 
y=−2x+4 
\begin{align*}y=14x21\end{align*}
y=14x−21 
\begin{align*}y=73x\end{align*}
y=7−3x
Find the intercepts of the following equations using the CoverUp Method.

\begin{align*}5x6y=15\end{align*}
5x−6y=15 
\begin{align*}3x4y=5\end{align*}
3x−4y=−5 
\begin{align*}2x+7y=11\end{align*}
2x+7y=−11 
\begin{align*}5x+10y=25\end{align*}
5x+10y=25  Do you prefer the Substitution Method or the CoverUp Method? Why?
In 13 – 24, use any method to find the intercepts and then graph the equation.

\begin{align*}y=2x+3\end{align*}
y=2x+3 
\begin{align*}6(x1)=2(y+3)\end{align*}
6(x−1)=2(y+3) 
\begin{align*}xy=5\end{align*}
x−y=5 
\begin{align*}x+y=8\end{align*}
x+y=8 
\begin{align*}4x+9y=0\end{align*}
4x+9y=0 
\begin{align*}\frac{1}{2} x+4y=12\end{align*}
12x+4y=12 
\begin{align*}x2y=4\end{align*}
x−2y=4 
\begin{align*}7x5y=10\end{align*}
7x−5y=10 
\begin{align*}4xy=3\end{align*}
4x−y=−3 
\begin{align*}xy=0\end{align*}
x−y=0 
\begin{align*}5x+y=5\end{align*}
5x+y=5 
\begin{align*}7x2y=6\end{align*}
7x−2y=−6  Which intercept does a vertical line have?
 Does the equation \begin{align*}y=5\end{align*}
y=5 have both an \begin{align*}x\end{align*}x− intercept and a \begin{align*}y\end{align*}y− intercept? Explain your answer.  Write an equation having only an \begin{align*}x\end{align*}
x− intercept at (–4, 0).  How many equations can be made with only one intercept at (0, 0)? Hint: Draw a picture to help you.
 What needs to be done to the following equation before you can use either method to find its intercepts? \begin{align*}3(x+2)=2(y+3)\end{align*}
3(x+2)=2(y+3)  At the local grocery store, strawberries cost $3.00 per pound and bananas cost $1.00 per pound. If I have $10 to spend between strawberries and bananas, draw a graph to show what combinations of each I can buy and spend exactly $10.
 A movie theater charges $7.50 for adult tickets and $4.50 for children. If the $900 theater takes in ticket sales for a particular screening, draw a graph that depicts the possibilities for the number of adult tickets and the number of child tickets sold.
 In football, touchdowns are worth 6 points, field goals are worth 3 points, and safeties are worth 2 points. Suppose there were no safeties and the team scored 36 points. Graph the situation to determine the combinations of field goals and touchdowns the team could have had.
Mixed Review
Determine whether each ordered pair is a solution to the equation.

\begin{align*}5x+2y=23;(7,6)\end{align*}
5x+2y=23;(7,−6) and (3, 4) 
\begin{align*}3a2b=6;(0,3)\end{align*}
3a−2b=6;(0,3) and \begin{align*}\left (\frac{5}{3},\frac{1}{2}\right )\end{align*}.  Graph the solutions to the equation \begin{align*}x=5\end{align*}.
 Solve: \begin{align*}\frac{4}{5} k16=\frac{1}{4}\end{align*}.
 Is the following relation a function? \begin{align*}\left \{(1,1),(0,0),(1,1),(2,3),(0,6)\right \}\end{align*}
 Using the number categories in Lesson 2.1, what is the best way to describe the domain of the following situation: The number of donuts purchased at a coffee shop on a particular day?
 Find the percent of change: Old price = $1,299; new price = $1,145.