# 11.2 Graphing linear equations

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Graph the equation $y=-1.$

## Solution

The equation $y=-1$ has only variable, $y.$ The value of $y$ is constant. All the ordered pairs in the table have the same $y$ -coordinate, $-1$ . We choose $0,3,$ and $-3$ as values for $x.$

$y=-1$
$x$ $y$ $\left(x,y\right)$
$-3$ $-1$ $\left(-3,-1\right)$
$0$ $-1$ $\left(0,-1\right)$
$3$ $-1$ $\left(3,-1\right)$

The graph is a horizontal line passing through the $y$ -axis at $–1$ as shown.

Graph the equation: $y=-4.$

Graph the equation: $y=3.$

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 $y=-3x$ and $y=-3$ in the same rectangular coordinate system.

## Solution

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: $y=-4x$ and $y=-4.$

Graph the equations in the same rectangular coordinate system: $y=3$ and $y=3x.$

## Key concepts

• Graph a linear equation by plotting points.
1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
2. Plot the points on a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
3. Draw the line through the points. Extend the line to fill the grid and put arrows on both ends of the line.
• Graph of a Linear Equation: The graph of a linear equation $ax+by=c$ is a straight line.
• Every point on the line is a solution of the equation.
• Every solution of this equation is a point on this line.
• A linear equation can be graphed by finding ordered pairs that represent solutions, plotting them on a coordinate grid, and drawing a line through them. See [link] .
• A linear equation forms a line when the solutions are plotted on a coordinate grid. All of the solutions are on the line, and any points that are not on the line are not solutions.
• A vertical line is a line that goes up and down on a coordinate grid. The $x\text{-coordinates}$ of a vertical line are all the same. See [link] .
• A horizontal line is a line that goes sideways on a coordinate grid. The $y\text{-coordinates}$ of a vertical line are all the same. See [link] .

## Practice makes perfect

Recognize the Relation Between the Solutions of an Equation and its Graph

For each ordered pair, decide

1. is the ordered pair a solution to the equation?
2. is the point on the line?

$y=x+2$

1. $\left(0,2\right)$
2. $\left(1,2\right)$
3. $\left(-1,1\right)$
4. $\left(-3,1\right)$

1. yes yes
2. no no
3. yes yes
4. yes yes

$y=x-4$

1. $\left(0,-4\right)$
2. $\left(3,-1\right)$
3. $\left(2,2\right)$
4. $\left(1,-5\right)$

1. yes yes
2. yes yes
3. no no
4. no no

$y=\frac{1}{2}x-3$

1. $\left(0,-3\right)$
2. $\left(2,-2\right)$
3. $\left(-2,-4\right)$
4. $\left(4,1\right)$

1. yes yes
2. yes yes
3. yes yes
4. no no

$y=\frac{1}{3}x+2$

1. $\left(0,2\right)$
2. $\left(3,3\right)$
3. $\left(-3,2\right)$
4. $\left(-6,0\right)$

1. yes yes
2. yes yes
3. no no
4. yes yes

Graph a Linear Equation by Plotting Points

In the following exercises, graph by plotting points.

$y=3x-1$

$y=2x+3$

$y=-2x+2$

$y=-3x+1$

$y=x+2$

$y=x-3$

$y=-x-3$

$y=-x-2$

$y=2x$

$y=3x$

$y=-4x$

$y=-2x$

$y=\frac{1}{2}x+2$

$y=\frac{1}{3}x-1$

$y=\frac{4}{3}x-5$

$y=\frac{3}{2}x-3$

$y=-\frac{2}{5}x+1$

$y=-\frac{4}{5}x-1$

$y=-\frac{3}{2}x+2$

$y=-\frac{5}{3}x+4$

$x+y=6$

$x+y=4$

$x+y=-3$

$x+y=-2$

$x-y=2$

$x-y=1$

$x-y=-1$

$x-y=-3$

$-x+y=4$

$-x+y=3$

$-x-y=5$

$-x-y=1$

$3x+y=7$

$5x+y=6$

$2x+y=-3$

$4x+y=-5$

$2x+3y=12$

$3x-4y=12$

$\frac{1}{3}x+y=2$

$\frac{1}{2}x+y=3$

Graph Vertical and Horizontal lines

In the following exercises, graph the vertical and horizontal lines.

$x=4$

$x=3$

$x=-2$

$x=-5$

$y=3$

$y=1$

$y=-5$

$y=-2$

$x=\frac{7}{3}$

$x=\frac{5}{4}$

In the following exercises, graph each pair of equations in the same rectangular coordinate system.

$y=-\frac{1}{2}x$ and $y=-\frac{1}{2}$

$y=-\frac{1}{3}x$ and $y=-\frac{1}{3}$

$y=2x$ and $y=2$

$y=5x$ and $y=5$

Mixed Practice

In the following exercises, graph each equation.

$y=4x$

$y=2x$

$y=-\frac{1}{2}x+3$

$y=\frac{1}{4}x-2$

$y=-x$

$y=x$

$x-y=3$

$x+y=-5$

$4x+y=2$

$2x+y=6$

$y=-1$

$y=5$

$2x+6y=12$

$5x+2y=10$

$x=3$

$x=-4$

## Everyday math

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 $400,800,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}1,200$ 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.

## Writing exercises

Explain how you would choose three $x\text{-values}$ to make a table to graph the line $y=\frac{1}{5}x-2.$

What is the difference between the equations of a vertical and a horizontal line?

## Self check

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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