11.3 Graphing with intercepts  (Page 2/4)

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Find the intercepts from an equation of a line

Recognizing that the $x\text{-intercept}$ occurs when $y$ is zero and that the $y\text{-intercept}$ occurs when $x$ is zero gives us a method to find the intercepts of a line from its equation. To find the $x\text{-intercept,}$ let $y=0$ and solve for $x.$ To find the $y\text{-intercept},$ let $x=0$ and solve for $y.$

Find the x And y From the equation of a line

Use the equation 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 .

x y
0
0

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

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

 Substitute 0 for y . Add. Divide by 2. The x -intercept is (3, 0).

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

 Substitute 0 for x . Multiply. Add. The y -intercept is (0, 6).

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

Find the intercepts: $3x+y=12$

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

Find the intercepts: $x+4y=8$

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

Find the intercepts of $4x-3y=12.$

Solution

To find the $x\text{-intercept,}$ let $y=0.$

 $4x-3y=12$ Distribute 0 for $y.$ $4x-3·0=12$ Multiply. $4x-0=12$ Subtract. $4x=12$ Divide by 4. $x=3$

The $x\text{-intercept}$ is $\left(3,0\right).$

To find the $y\text{-intercept},$ let $x=0.$

 $4x-3y=12$ Substitute 0 for $x.$ $4·0-3y=12$ Multiply. $0-3y=12$ Simplify. $-3y=12$ Divide by −3. $y=-4$

The $y\text{-intercept}$ is $\left(0,-4\right).$

The intercepts are the points $\left(-3,0\right)$ and $\left(0,-4\right).$

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

Find the intercepts of the line: $3x-4y=12.$

x -intercept (4,0); y -intercept: (0,−3)

Find the intercepts of the line: $2x-4y=8.$

x -intercept (4,0); y -intercept: (0,−2)

Graph a line using the intercepts

To graph a linear equation by plotting points, you can use the intercepts as two of your three points. Find the two intercepts, and then a third point to ensure accuracy, and draw the line. This method is often the quickest way to graph a line.

Graph $-x+2y=6$ using intercepts.

Solution

First, find the $x\text{-intercept}.$ Let $y=0,$

$\begin{array}{}\\ \phantom{\rule{0.7em}{0ex}}-x+2y=6\\ -x+2\left(0\right)=6\\ \phantom{\rule{2.8em}{0ex}}-x=6\\ \phantom{\rule{4.3em}{0ex}}x=-6\end{array}$

The $x\text{-intercept}$ is $\left(–6,0\right).$

Now find the $y\text{-intercept}.$ Let $x=0.$

$\begin{array}{}\\ -x+2y=6\\ -0+2y=6\\ \\ \\ \phantom{\rule{2.4em}{0ex}}2y=6\\ \phantom{\rule{3em}{0ex}}y=3\end{array}$

The $y\text{-intercept}$ is $\left(0,3\right).$

Find a third point. We’ll use $x=2,$

$\begin{array}{}\\ -x+2y=6\\ -2+2y=6\\ \\ \\ \phantom{\rule{2.4em}{0ex}}2y=8\\ \phantom{\rule{3em}{0ex}}y=4\end{array}$

A third solution to the equation is $\left(2,4\right).$

Summarize the three points in a table and then plot them on a graph.

 $-x+2y=6$ x y (x,y) $2$ $4$ $\left(2,4\right)$ $0$ $3$ $\left(0,3\right)$ $2$ $4$ $\left(2,4\right)$

Do the points line up? Yes, so draw line through the points.

Graph the line using the intercepts: $x-2y=4.$

Graph the line using the intercepts: $-x+3y=6.$

Graph a line using the intercepts.

1. Find the $x-$ and $\text{y-intercepts}$ 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.

Graph $4x-3y=12$ using intercepts.

Solution

Find the intercepts and a third point.

We list the points and show the graph.

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

Graph the line using the intercepts: $5x-2y=10.$

Graph the line using the intercepts: $3x-4y=12.$

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 intercepts are the same point, we need two more points to graph the line. As always, we can choose any values for $x,$ so we’ll let $x$ be $1$ and $-1.$

Organize the points in a table.

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

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

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