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Use the equation of the line. To find:
Find the intercepts of $2x+y=6$ .
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 $4x\u20133y=12$ .
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 | $\mathrm{-4}$ |
Find the intercepts of $3x\u20134y=12.$
x - intercept: $\left(4,0\right)$ , y - intercept: $\left(0,\mathrm{-3}\right)$
Find the intercepts of $2x\u20134y=8.$
x - intercept: $\left(4,0\right)$ , y - intercept: $\left(0,\mathrm{-2}\right)$
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.
The steps to graph a linear equation using the intercepts are summarized below.
Graph $4x\u20133y=12$ using the intercepts.
Find the intercepts and a third point.
We list the points in [link] and show the graph below.
$4x-3y=12$ | ||
$x$ | $y$ | $(x,y)$ |
3 | 0 | $(3,0)$ |
0 | $\mathrm{-4}$ | $(0,\mathrm{-4})$ |
6 | 4 | $(6,4)$ |
Graph $y=5x$ using the intercepts.
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.
See [link] .
$y=5x$ | ||
$x$ | $y$ | $(x,y)$ |
0 | 0 | $(0,0)$ |
1 | 5 | $(1,5)$ |
$\mathrm{-1}$ | $\mathrm{-5}$ | $(\mathrm{-1},\mathrm{-5})$ |
Plot the three points, check that they line up, and draw the line.
Identify the x - and y - Intercepts on a Graph
In the following exercises, find the x - and y - intercepts on each graph.
Find the x - and y - Intercepts from an Equation of a Line
In the following exercises, find the intercepts for each equation.
$x+y=\mathrm{-2}$
$\left(\mathrm{-2},0\right),\left(0,\mathrm{-2}\right)$
$x+y=\mathrm{-5}$
$x\u2013y=1$
$x\u2013y=\mathrm{-3}$
$\left(\mathrm{-3},0\right),\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}\left(0,3\right)$
$x\u2013y=\mathrm{-4}$
$x+2y=10$
$x\u20133y=12$
$\left(12,0\right),\left(0,\mathrm{-4}\right)$
$x\u20132y=8$
$5x\u2013y=5$
$2x+3y=6$
$3x\u20132y=12$
$\left(4,0\right),\left(0,\mathrm{-6}\right)$
$3x\u20135y=30$
$y=\frac{1}{3}x+1$
$\left(3,0\right),\left(0,\mathrm{-1}\right)$
$y=\frac{1}{4}x-1$
$y=\frac{1}{5}x+2$
$\left(\mathrm{-10},0\right),\left(0,2\right)$
$y=\frac{1}{3}x+4$
$y=\mathrm{-2}x$
Graph a Line Using the Intercepts
In the following exercises, graph using the intercepts.
$\u2013x+4y=8$
$x+y=\mathrm{-1}$
$x\u2013y=2$
$x\u2013y=\mathrm{-3}$
$3x+2y=12$
$5x\u20132y=10$
$3x\u20134y=\mathrm{-12}$
$2x\u2013y=\mathrm{-8}$
$y=\mathrm{-4}x$
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.
ⓐ
$\left(0,1000\right),\left(15,0\right)$
ⓑ 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.
How do you find the x - intercept of the graph of $3x\u20132y=6$ ?
Answers will vary.
Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation $4x+y=\mathrm{-4}$ ? 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?
Answers will vary.
Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation $y=6$ ? Why?
ⓐ 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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