4.3 Graph with intercepts  (Page 2/5)

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Find the x - and y - intercepts from the equation of a line

Use the equation of the line. To find:

• the x - intercept of the line, let $y=0$ and solve for $x$ .
• the y - intercept of the line, let $x=0$ and solve for $y$ .

Find the intercepts of $2x+y=6$ .

Solution

We will let $y=0$ to find the x - intercept, and let $x=0$ to find the y - intercept. We will fill in the table, which reminds us of what we need to find.

To find the x - intercept, let $y=0$ .

 Let y = 0. Simplify. The x -intercept is (3, 0) To find the y -intercept, let x = 0. Let x = 0. Simplify. The y -intercept is (0, 6)

The intercepts are the points $\left(3,0\right)$ and $\left(0,6\right)$ as shown in [link] .

 $2x+y=6$ $x$ $y$ 3 0 0 6

Find the intercepts of $3x+y=12.$

x - intercept: $\left(4,0\right)$ , y - intercept: $\left(0,12\right)$

Find the intercepts of $x+4y=8.$

x - intercept: $\left(8,0\right)$ , y - intercept: $\left(0,2\right)$

Find the intercepts of $4x–3y=12$ .

Solution

 To find the x -intercept, let y = 0. Let y = 0. Simplify. The x -intercept is (3, 0) To find the y -intercept, let x = 0. Let x = 0. Simplify. The y -intercept is (0, −4)

The intercepts are the points (3, 0) and (0, −4) as shown in [link] .

 $4x-3y=12$ $x$ $y$ 3 0 0 $-4$

Find the intercepts of $3x–4y=12.$

x - intercept: $\left(4,0\right)$ , y - intercept: $\left(0,-3\right)$

Find the intercepts of $2x–4y=8.$

x - intercept: $\left(4,0\right)$ , y - intercept: $\left(0,-2\right)$

Graph a line using the intercepts

To graph a linear equation by plotting points, you need to find three points whose coordinates are solutions to the equation. You can use the x - and y - intercepts as two of your three points. Find the intercepts, and then find a third point to ensure accuracy. Make sure the points line up—then draw the line. This method is often the quickest way to graph a line.

How to graph a line using intercepts

Graph $–x+2y=6$ using the intercepts.

Solution

Graph $x–2y=4$ using the intercepts.

Graph $–x+3y=6$ using the intercepts.

The steps to graph a linear equation using the intercepts are summarized below.

Graph a linear equation using the intercepts.

1. Find the x - and y - intercepts of the line.
• Let $y=0$ and solve for $x$
• Let $x=0$ and solve for $y$ .
2. Find a third solution to the equation.
3. Plot the three points and check that they line up.
4. Draw the line.

Graph $4x–3y=12$ using the intercepts.

Solution

Find the intercepts and a third point.

We list the points in [link] and show the graph below.

 $4x-3y=12$ $x$ $y$ $\left(x,y\right)$ 3 0 $\left(3,0\right)$ 0 $-4$ $\left(0,-4\right)$ 6 4 $\left(6,4\right)$

Graph $5x–2y=10$ using the intercepts.

Graph $3x–4y=12$ using the intercepts.

Graph $y=5x$ using the intercepts.

Solution

This line has only one intercept. It is the point $\left(0,0\right)$ .

To ensure accuracy we need to plot three points. Since the x - and y - intercepts are the same point, we need two more points to graph the line.

 $y=5x$ $x$ $y$ $\left(x,y\right)$ 0 0 $\left(0,0\right)$ 1 5 $\left(1,5\right)$ $-1$ $-5$ $\left(-1,-5\right)$

Plot the three points, check that they line up, and draw the line.

Graph $y=4x$ using the intercepts.

Graph $y=\text{−}x$ the intercepts.

Key concepts

• Find the x - and y - Intercepts from the Equation of a Line
• Use the equation of the line to find the x - intercept of the line, let $y=0$ and solve for x .
• Use the equation of the line to find the y - intercept of the line, let $x=0$ and solve for y .
• Graph a Linear Equation using the Intercepts
1. Find the x - and y - intercepts of the line.
Let $y=0$ and solve for x .
Let $x=0$ and solve for y .
2. Find a third solution to the equation.
3. Plot the three points and then check that they line up.
4. Draw the line.

• Strategy for Choosing the Most Convenient Method to Graph a Line:
• Consider the form of the equation.
• If it only has one variable, it is a vertical or horizontal line.
$x=a$ is a vertical line passing through the x - axis at $a$
$y=b$ is a horizontal line passing through the y - axis at $b$ .
• If y is isolated on one side of the equation, graph by plotting points.
• Choose any three values for x and then solve for the corresponding y - values.
• If the equation is of the form $ax+by=c$ , find the intercepts. Find the x - and y - intercepts and then a third point.

Practice makes perfect

Identify the x - and y - Intercepts on a Graph

In the following exercises, find the x - and y - intercepts on each graph.

$\left(3,0\right),\left(0,3\right)$

$\left(5,0\right),\left(0,-5\right)$

$\left(-2,0\right),\left(0,-2\right)$

$\left(-1,0\right),\left(0,1\right)$

$\left(6,0\right),\left(0,3\right)$

$\left(0,0\right)$

Find the x - and y - Intercepts from an Equation of a Line

In the following exercises, find the intercepts for each equation.

$x+y=4$

$\left(4,0\right),\left(0,4\right)$

$x+y=3$

$x+y=-2$

$\left(-2,0\right),\left(0,-2\right)$

$x+y=-5$

$x–y=5$

$\left(5,0\right),\left(0,-5\right)$

$x–y=1$

$x–y=-3$

$\left(-3,0\right),\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}\left(0,3\right)$

$x–y=-4$

$x+2y=8$

$\left(8,0\right),\left(0,4\right)$

$x+2y=10$

$3x+y=6$

$\left(2,0\right),\left(0,6\right)$

$3x+y=9$

$x–3y=12$

$\left(12,0\right),\left(0,-4\right)$

$x–2y=8$

$4x–y=8$

$\left(2,0\right),\left(0,-8\right)$

$5x–y=5$

$2x+5y=10$

$\left(5,0\right),\left(0,2\right)$

$2x+3y=6$

$3x–2y=12$

$\left(4,0\right),\left(0,-6\right)$

$3x–5y=30$

$y=\frac{1}{3}x+1$

$\left(3,0\right),\left(0,-1\right)$

$y=\frac{1}{4}x-1$

$y=\frac{1}{5}x+2$

$\left(-10,0\right),\left(0,2\right)$

$y=\frac{1}{3}x+4$

$y=3x$

$\left(0,0\right)$

$y=-2x$

$y=-4x$

$\left(0,0\right)$

$y=5x$

Graph a Line Using the Intercepts

In the following exercises, graph using the intercepts.

$–x+5y=10$

$–x+4y=8$

$x+2y=4$

$x+2y=6$

$x+y=2$

$x+y=5$

$x+y=-3$

$x+y=-1$

$x–y=1$

$x–y=2$

$x–y=-4$

$x–y=-3$

$4x+y=4$

$3x+y=3$

$2x+4y=12$

$3x+2y=12$

$3x–2y=6$

$5x–2y=10$

$2x–5y=-20$

$3x–4y=-12$

$3x–y=-6$

$2x–y=-8$

$y=-2x$

$y=-4x$

$y=x$

$y=3x$

Everyday math

Road trip. Damien is driving from Chicago to Denver, a distance of 1000 miles. The x - axis on the graph below shows the time in hours since Damien left Chicago. The y - axis represents the distance he has left to drive.

1. Find the x - and y - intercepts.
2. Explain what the x - and y - intercepts mean for Damien.

$\left(0,1000\right),\left(15,0\right)$
At $\left(0,1000\right)$ , he has been gone 0 hours and has 1000 miles left. At $\left(15,0\right)$ , he has been gone 15 hours and has 0 miles left to go.

Road trip. Ozzie filled up the gas tank of his truck and headed out on a road trip. The x - axis on the graph below shows the number of miles Ozzie drove since filling up. The y - axis represents the number of gallons of gas in the truck’s gas tank.

1. Find the x - and y - intercepts.
2. Explain what the x - and y - intercepts mean for Ozzie.

Writing exercises

How do you find the x - intercept of the graph of $3x–2y=6$ ?

Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation $4x+y=-4$ ? Why?

Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation $y=\frac{2}{3}x-2$ ? Why?

Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation $y=6$ ? Why?

Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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