4.5: Intercepts by Substitution
Suppose the number of gallons of gas left in the tank of a car depends on the number of miles driven and can be represented by a linear equation. You know how much gas is in the tank when 0 miles have been driven, and you also know after how many miles you will have 0 gallons left in the tank. Could you graph the linear equation? In this Concept, you'll learn how to graph equations in situations like this one, where you know the intercepts.
Guidance
As you may have seen in the previous Concept, graphing solutions to an equation of two variables can be time-consuming. Fortunately, there are several ways to make graphing solutions easier. This Concept will focus on graphing a line by finding its intercepts. A future Concept will show you how to graph a line using its slope and \begin{align*}y-\end{align*}intercept.
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 \begin{align*}x-\end{align*}intercept is an ordered pair at which the line crosses the \begin{align*}x-\end{align*}axis (the horizontal axis). Its ordered pair has the form \begin{align*}(x,0)\end{align*}.
The \begin{align*}y-\end{align*}intercept is an ordered pair at which the line crosses the \begin{align*}y-\end{align*}axis (the vertical axis). Its ordered pair has the form \begin{align*}(0,y)\end{align*}.
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 A
Graph \begin{align*}2x+3y=-6\end{align*} using its intercepts.
Solution: The \begin{align*}x-\end{align*}intercept has the ordered pair \begin{align*}(x,0)\end{align*}. Therefore, the \begin{align*}y-\end{align*}coordinate has a value of zero. By substituting zero for the variable of \begin{align*}y\end{align*}, the equation becomes:
\begin{align*}2x+3(0)=-6\end{align*}
Continue solving for \begin{align*}x\end{align*}:
\begin{align*}2x+0& =-6\\ 2x& =-6\\ x& =-3\end{align*}
The \begin{align*}x-\end{align*}intercept has an ordered pair of (–3, 0).
Repeat the process to find the \begin{align*}y-\end{align*}intercept. The ordered pair of the \begin{align*}y-\end{align*}intercept is \begin{align*}(0,y)\end{align*}. Using substitution:
\begin{align*}2(0)+3y& =-6\\ 3y& =-6\\ y& =-2\end{align*}
The \begin{align*}y-\end{align*}intercept has the ordered pair (0, –2).
To graph the line formed by the solutions of the equation \begin{align*}2x+3y=-6\end{align*}, graph the two intercepts and connect them with a straight line.
Example B
Graph \begin{align*}4x-2y=8\end{align*} using its intercepts.
Solution: Determine the \begin{align*}x-\end{align*}intercept by substituting zero for the variable \begin{align*}y\end{align*}.
\begin{align*}4x-2(0)& =8\\ 4x& =8\\ x& =2\end{align*}
The ordered pair of the \begin{align*}x-\end{align*}intercept is (2, 0). By repeating this process, you'll find the \begin{align*}y-\end{align*}intercept has the ordered pair (0, –4). Graph these two ordered pairs and connect them with a line.
Example C
Graph \begin{align*}x-2y=10\end{align*} using intercepts.
Solution:
First, find the intercepts by substituting in zero for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}:
\begin{align*} \text{Substitute in zero for x.} && (0)-2y &=10 \\ \text{Simplify.} && -2y&=10 \\ \text{Solve for y.}&& y&=-5 \end{align*}
This means that the \begin{align*}y\end{align*}-intercept is at the point (0,-5).
\begin{align*} \text{Substitute in zero for y.} && x-6(0) &=10 \\ \text{Solve for x.}&& x&=10 \end{align*}
This means that the \begin{align*}x\end{align*}-intercept is at the point (10,0).
Guided Practice
Find the intercepts and use them to graph \begin{align*}y=-2x+8\end{align*}.
Solution:
Substitute in zero for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}, in order to find the intercepts.
\begin{align*} \text{Substitute in zero for x.} && y&=-2(0)+8 \\ \text{Solve for y.}&& y&=8 \end{align*}
This means that the \begin{align*}y\end{align*}-intercept is at the point (0,8).
\begin{align*} \text{Substitute in zero for y.} && 0&=-2x+8 \\ \text{Solve for x.}&& x&=4 \end{align*}
This means that the \begin{align*}x\end{align*}-intercept is at the point (4,0).
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: Graphing Using Intercepts (12:18)
- Define intercept.
- What is the ordered pair for an \begin{align*}x-\end{align*}intercept?
Find the intercepts for the following equations using substitution.
- \begin{align*}y=3x-6\end{align*}
- \begin{align*}y=-2x+4\end{align*}
- \begin{align*}y=14x-21\end{align*}
- \begin{align*}y=7-3x\end{align*}
- Which intercept does a vertical line have?
- Does the equation \begin{align*}y=5\end{align*} have both an \begin{align*}x-\end{align*}intercept and a \begin{align*}y-\end{align*}intercept? Explain your answer.
- Write an equation having only an \begin{align*}x-\end{align*}intercept at (–4, 0).
- How many equations can be made with only one intercept at (0, 0)? Hint: Draw a picture to help you.
Mixed Review
For 11-12, Determine whether each ordered pair is a solution to the equation.
- \begin{align*}5x+2y=23;(7,-6)\end{align*} and (3, 4)
- \begin{align*}3a-2b=6;(0,3)\end{align*} 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} k-16=-\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 either a finite set of counting numbers, whole numbers, or integers, how would you 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.
-intercept
An ordered pair at which the line crosses the axis (the horizontal axis). Its ordered pair has the form .-intercept
The intercept is an ordered pair at which the line crosses the axis (the vertical axis). Its ordered pair has the form .Intercept
The intercepts of a curve are the locations where the curve intersects the and axes. An intercept is a point at which the curve intersects the -axis. A intercept is a point at which the curve intersects the -axis.Intercept Method
The intercept method is a way of graphing a linear function by using the coordinates of the and -intercepts. The graph is drawn by plotting these coordinates on the Cartesian plane and then joining them with a straight line.Image Attributions
Here you'll learn how to use substitution to find the intercepts of a line so that you can graph a linear equation.
Concept Nodes:
-intercept
An ordered pair at which the line crosses the axis (the horizontal axis). Its ordered pair has the form .-intercept
The intercept is an ordered pair at which the line crosses the axis (the vertical axis). Its ordered pair has the form .Intercept
The intercepts of a curve are the locations where the curve intersects the and axes. An intercept is a point at which the curve intersects the -axis. A intercept is a point at which the curve intersects the -axis.Intercept Method
The intercept method is a way of graphing a linear function by using the coordinates of the and -intercepts. The graph is drawn by plotting these coordinates on the Cartesian plane and then joining them with a straight line.