Graphs with Intercepts

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.