Graphing 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.
Example
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) |
| \(-6\) | \(0\) | \(\left(-6,0\right)\) |
| \(0\) | \(3\) | \(\left(0,3\right)\) |
| \(2\) | \(4\) | \(\left(2,4\right)\) |
Do the points line up? Yes, so draw line through the points.
How to Graph a line using the intercepts.
- Find the \(x-\) and \(\text{y-intercepts}\) of the line.
- Let \(y=0\) and solve for \(x\)
- Let \(x=0\) and solve for \(y.\)
- Find a third solution to the equation.
- Plot the three points and then check that they line up.
- Draw the line.
Example
Graph \(4x-3y=12\) using intercepts.
Solution
Find the intercepts and a third point.
We list the points and show the graph.
| \(4x-3y=12\) | ||
|---|---|---|
| \(x\) | \(y\) | \(\left(x,y\right)\) |
| \(3\) | \(0\) | \(\left(3,0\right)\) |
| \(0\) | \(-4\) | \(\left(0,-4\right)\) |
| \(6\) | \(4\) | \(\left(6,4\right)\) |
Example
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.