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While we could graph any linear equation by plotting points, it may not always be the most convenient method. This table shows six of equations we’ve graphed in this chapter, and the methods we used to graph them.
Equation | Method | |
---|---|---|
#1 | $y=2x+1$ | Plotting points |
#2 | $y=\frac{1}{2}x+3$ | Plotting points |
#3 | $x=\mathrm{-7}$ | Vertical line |
#4 | $y=4$ | Horizontal line |
#5 | $2x+y=6$ | Intercepts |
#6 | $4x-3y=12$ | Intercepts |
What is it about the form of equation that can help us choose the most convenient method to graph its line?
Notice that in equations #1 and #2, y is isolated on one side of the equation, and its coefficient is 1. We found points by substituting values for x on the right side of the equation and then simplifying to get the corresponding y- values.
Equations #3 and #4 each have just one variable. Remember, in this kind of equation the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines.
In equations #5 and #6, both x and y are on the same side of the equation. These two equations are of the form $Ax+By=C$ . We substituted $y=0$ and $x=0$ to find the x- and y- intercepts, and then found a third point by choosing a value for x or y .
This leads to the following strategy for choosing the most convenient method to graph a line.
Identify the most convenient method to graph each line:
ⓐ $\phantom{\rule{0.2em}{0ex}}y=\mathrm{-3}$
This equation has only one variable, $y.$ Its graph is a horizontal line crossing the $y\text{-axis}$ at $\mathrm{-3}.$
ⓑ $\phantom{\rule{0.2em}{0ex}}4x\mathrm{-6}y=12$
This equation is of the form $Ax+By=C.$ Find the intercepts and one more point.
ⓒ $\phantom{\rule{0.2em}{0ex}}x=2$
There is only one variable, $x.$ The graph is a vertical line crossing the $x\text{-axis}$ at $2.$
ⓓ $\phantom{\rule{0.2em}{0ex}}y=\frac{2}{5}x\mathrm{-1}$
Since $y$ is isolated on the left side of the equation, it will be easiest to graph this line by plotting three points.
Identify the most convenient method to graph each line:
Identify the most convenient method to graph each line:
x | y |
---|---|
0 | |
0 |
Identify the Intercepts on a Graph
In the following exercises, find the $x\text{-}$ and $y\text{-}$ intercepts.
Find the $x$ and $y$ Intercepts from an Equation of a Line
In the following exercises, find the intercepts.
$x+y=\mathrm{-5}$
$x-y=\mathrm{-4}$
$x+2y=10$
$x\mathrm{-2}y=8$
$2x+3y=6$
$3x\mathrm{-5}y=30$
$y=\frac{1}{4}x\mathrm{-1}$
$y=\frac{1}{3}x+4$
$y=\mathrm{-2}x$
Graph a Line Using the Intercepts
In the following exercises, graph using the intercepts.
$-x+4y=8$
$x+y=\mathrm{-1}$
$x-y=\mathrm{-3}$
$2x-y=\mathrm{-8}$
$3x+2y=12$
$5x\mathrm{-2}y=10$
$3x\mathrm{-4}y=\mathrm{-12}$
$y=\mathrm{-4}x$
Choose the Most Convenient Method to Graph a Line
In the following exercises, identify the most convenient method to graph each line.
$x=\mathrm{-3}$
$y=\mathrm{-5}x+2$
$y=\frac{4}{5}x\mathrm{-3}$
$y=\mathrm{-1}$
$2x\mathrm{-5}y=\mathrm{-10}$
$y=-\frac{1}{3}x+5$
Road trip Damien is driving from Chicago to Denver, a distance of $\mathrm{1,000}$ miles. The $x\text{-axis}$ on the graph below shows the time in hours since Damien left Chicago. The $y\text{-axis}$ represents the distance he has left to drive.
ⓐ Find the $x\text{-}$ and $y\text{-}$ intercepts
ⓑ Explain what the $x\text{-}$ and $y\text{-}$ intercepts mean for Damien.
ⓐ (0,1,000),(15,0). ⓑ At (0,1,000) he left Chicago 0 hours ago and has 1,000 miles left to drive. At (15,0) he left Chicago 15 hours ago and has 0 miles left to drive.
Road trip Ozzie filled up the gas tank of his truck and went on a road trip. The $x\text{-axis}$ on the graph shows the number of miles Ozzie drove since filling up. The $y\text{-axis}$ represents the number of gallons of gas in the truck’s gas tank.
ⓐ Find the $x\text{-}$ and $y\text{-}$ intercepts.
ⓑ Explain what the $x\text{-}$ and $y\text{-}$ intercepts mean for Ozzie.
How do you find the $x\text{-intercept}$ of the graph of $3x\mathrm{-2}y=6?$
Answers will vary.
How do you find the $y\text{-intercept}$ of the graph of $5x-y=10?$
Do you prefer to graph the equation $4x+y=\mathrm{-4}$ by plotting points or intercepts? Why?
Answers will vary.
Do you prefer to graph the equation $y=\frac{2}{3}x\mathrm{-2}$ by plotting points or intercepts? 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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