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Graph the equation $y=\mathrm{-1}.$
The equation $y=\mathrm{-1}$ has only variable, $y.$ The value of $y$ is constant. All the ordered pairs in the table have the same $y$ -coordinate, $\mathrm{-1}$ . We choose $0,3,$ and $\mathrm{-3}$ as values for $x.$
$y=\mathrm{-1}$ | ||
---|---|---|
$x$ | $y$ | $(x,y)$ |
$\mathrm{-3}$ | $\mathrm{-1}$ | $(\mathrm{-3},\mathrm{-1})$ |
$0$ | $\mathrm{-1}$ | $(0,\mathrm{-1})$ |
$3$ | $\mathrm{-1}$ | $(3,\mathrm{-1})$ |
The graph is a horizontal line passing through the $y$ -axis at $\mathrm{\u20131}$ as shown.
The equations for vertical and horizontal lines look very similar to equations like $y=4x.$ What is the difference between the equations $y=4x$ and $y=4?$
The equation $y=4x$ has both $x$ and $y.$ The value of $y$ depends on the value of $x.$ The $y\text{-coordinate}$ changes according to the value of $x.$
The equation $y=4$ has only one variable. The value of $y$ is constant. The $y\text{-coordinate}$ is always $4.$ It does not depend on the value of $x.$
The graph shows both equations.
Notice that the equation $y=4x$ gives a slanted line whereas $y=4$ gives a horizontal line.
Graph $y=\mathrm{-3}x$ and $y=\mathrm{-3}$ in the same rectangular coordinate system.
Find three solutions for each equation. Notice that the first equation has the variable
$x,$ while the second does not. Solutions for both equations are listed.
The graph shows both equations.
Graph the equations in the same rectangular coordinate system: $y=\mathrm{-4}x$ and $y=\mathrm{-4}.$
Graph the equations in the same rectangular coordinate system: $y=3$ and $y=3x.$
Recognize the Relation Between the Solutions of an Equation and its Graph
For each ordered pair, decide
$y=x+2$
$y=x-4$
$y=\frac{1}{2}x-3$
$y=\frac{1}{3}x+2$
Graph a Linear Equation by Plotting Points
In the following exercises, graph by plotting points.
$y=\mathrm{-3}x+1$
$y=\mathrm{-2}x$
$y=\frac{1}{3}x-1$
$y=\frac{3}{2}x-3$
$y=-\frac{4}{5}x-1$
$y=-\frac{5}{3}x+4$
$x+y=\mathrm{-2}$
$x-y=\mathrm{-3}$
$4x+y=\mathrm{-5}$
$3x-4y=12$
$\frac{1}{2}x+y=3$
Graph Vertical and Horizontal lines
In the following exercises, graph the vertical and horizontal lines.
$x=\mathrm{-5}$
$y=\mathrm{-2}$
$x=\frac{5}{4}$
In the following exercises, graph each pair of equations in the same rectangular coordinate system.
$y=5x$ and $y=5$
Mixed Practice
In the following exercises, graph each equation.
Motor home cost The Robinsons rented a motor home for one week to go on vacation. It cost them $\text{\$594}$ plus $\text{\$0.32}$ per mile to rent the motor home, so the linear equation $y=594+0.32x$ gives the cost, $y,$ for driving $x$ miles. Calculate the rental cost for driving $400,800,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\mathrm{1,200}$ miles, and then graph the line.
$722, $850, $978
Weekly earning At the art gallery where he works, Salvador gets paid $\text{\$200}$ per week plus $\text{15\%}$ of the sales he makes, so the equation $y=200+0.15x$ gives the amount $y$ he earns for selling $x$ dollars of artwork. Calculate the amount Salvador earns for selling $\text{\$900, \$1,600},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{\$2,000},$ and then graph the line.
Explain how you would choose three $x\text{-values}$ to make a table to graph the line $y=\frac{1}{5}x-2.$
Answers will vary.
What is the difference between the equations of a vertical and a horizontal line?
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?
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