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.

Example 4.22

Graph $$–x+2y=6$$ using the intercepts.

Solution

Step 1. Find the $x$ - and $y$ intercepts of the line.

Let $y=0$ and solve for $x$.

Let $x=0$ and solve for $y$.

Find the $x$ -intercept.

Find the $y$ -intercept.

$\begin{align} \begin{array}{r} \text { Let } y=0 \\ -x+2 y=6 \\ -x+2(0)=6 \\ -x=6 \\ x=-6 \end{array} \end{align}$
The $x$ -intercept is $(-6,0)$.

$\begin{align} \begin{array}{r} \text { Let } x=0 \\ -x+2 y=6 \\ -0+2 y=6 \\ 2 y=6 \\ y=3 \end{array} \end{align}$

The $y$ -intercept is $(0,3)$

Step 2. Find another solution to the equation.

We'll use $x=2$.

$\begin{align} \begin{array}{r} \text { Let } x=2 \\ -x+2 y=6 \\ -2+2 y=6 \\ 2 y=8 \\ y=4 \end{array} \end{align}$

A third point is $(2,4)$.

Step 3. Plot the three points. Check that the points line up.

$x$	$y$	$(x.y)$
$-6$	$0$	$(-6, 0)$
$0$	$3$	$(0. 3)$
$2$	$4$	$(2. 4)$

Step 3. Plot the three points. Check that the points line up.

Step 4. Draw the line.

See the graph.

Step 4. Draw the line.

Try It 4.43

Graph $x–2y=4$ using the intercepts.

Try It 4.44

Graph $–x+3y=6$ using the intercepts.

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

HOW TO

Graph a linear equation using the intercepts.

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

Example 4.23

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

Solution

Find the intercepts and a third point.

\[x \text { -intercept, let } y=0\]	\[y \text { -intercept, let } x=0\]	\[\text { third point, let } y=4\]
$\begin{aligned} 4 x-3 y &=12 \\ 4 x-3(0) &=12 \\ 4 x &=12 \\ x &=3 \end{aligned}$	\[\begin{array}{r} 4 x-3 y=12 \\ 4(0)-3 y=12 \\ -3 y=12 \\ y=-4 \end{array} \]	\[\begin{array}{r} 4 x-3 y=12 \\ 4 x-3(4)=12 \\ 4 x-12=12 \\ 4 x=24 \\ x=6\end{array}\]

We list the points in Table 4.27 and show the graph below.

$4x−3y=12$
$x$	$y$	$(x,y)$
$3$	$0$	$(3,0)$
$0$	−$4$	$(0,−4)$
$6$	$4$	$(6,4)$

Table 4.27

The figure shows the graph of a straight line going through three points on the x y- coordinate plane. The x- axis of the pla

Try It 4.45

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

Try It 4.46

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

Example 4.24

Graph $$y=5x$$ using the intercepts.

Solution

$$x$$-intercept	$$y$$-intercept
Let $$y=0$$.	Let $$x=0$$.
$$y=5 x$$	$$y=5 x$$
$$0=5 x$$	$$y=5 \cdot 0$$
$$ 0=x$$	$$y=0$$
$$ (0,0) $$	$$ (0,0)$$

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

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.

Let $$x=1$$.	Let $$x=-1$$.
$$y=5 x$$	$$y=5 x$$
$$y=5 \cdot 1$$	$$y=5 (-1)$$
$$y=5$$	$$y=-5$$

See Table 4.28.

y=5x
x	y	(x,y)
$0$	$0$	$(0,0)$
$1$	$5$	$(1,5)$
$−1$	$−5$	$(−1,−5)$

Table 4.28

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

The figure shows the graph of a straight line going through three points on the x y- coordinate plane. The x- axis of the pla

Try It 4.47

Graph $y=4x$ using the intercepts.

Try It 4.48

Graph $y=−x$ the intercepts.

\(4x−3y=12\)
\(x\)	\(y\)	\((x,y)\)
\(3\)	\(0\)	\((3,0)\)
\(0\)	−\(4\)	\((0,−4)\)
\(6\)	\(4\)	\((6,4)\)

y=5x
x	y	(x,y)
\(0\)	\(0\)	\((0,0)\)
\(1\)	\(5\)	\((1,5)\)
\(−1\)	\(−5\)	\((−1,−5)\)