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?

### Finding Intercepts by Using Substitution

Graphing solutions to an equation of two variables using a table 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.

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 **intercept** is an ordered pair at which the line crosses the

The **intercept** is an ordered pair at which the line crosses the

By finding the intercepts of an equation, you can quickly graph all the possible solutions to the equation.

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.

#### Let's graph the following linear equations using its intercepts:

2x+3y=−6

The

Continue solving for

The

Repeat the process to find the

The

To graph the line formed by the solutions of the equation

4x−2y=8

Determine the

The ordered pair of the

x−2y=10

First, find the intercepts by substituting in zero for

This means that the

This means that the

### Examples

#### Example 1

Earlier, you were asked to suppose that the number of gallons left in the tank of a car depends on the number of miles driven and can be represented by a linear equation. Could you graph the linear equation if you know how much gas is in the tank when 0 miles have been driven and how many miles it will take to have 0 gallons left in the tank?

Yes, you can draw a line with any two points. In this case, the amount of gas in the tank when 0 miles have been driven will be the

#### Example 2

Find the intercepts and use them to graph

Substitute in zero for

This means that the

This means that the

### Review

- Define intercept.
- What is the ordered pair for an
x− intercept?

Find the intercepts for the following equations using substitution.

y=3x−6 y=−2x+4 y=14x−21 y=7−3x

- Which intercept does a vertical line have?
- Does the equation
y=5 have both anx− 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.

**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*}
5x+2y=23;(7,−6) and (3, 4) - \begin{align*}3a-2b=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} 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.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 4.5.