# 11.2 Graphing linear equations  (Page 4/6)

 Page 4 / 6

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 $y=\frac{1}{5}x-2.$ Answers will vary. 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? #### Questions & Answers Is there any normative that regulates the use of silver nanoparticles? Damian Reply what king of growth are you checking .? Renato What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ? Stoney Reply why we need to study biomolecules, molecular biology in nanotechnology? Adin Reply ? Kyle yes I'm doing my masters in nanotechnology, we are being studying all these domains as well.. Adin why? Adin what school? Kyle biomolecules are e building blocks of every organics and inorganic materials. Joe anyone know any internet site where one can find nanotechnology papers? Damian Reply research.net kanaga sciencedirect big data base Ernesto Introduction about quantum dots in nanotechnology Praveena Reply what does nano mean? Anassong Reply nano basically means 10^(-9). nanometer is a unit to measure length. Bharti do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment? Damian Reply absolutely yes Daniel how to know photocatalytic properties of tio2 nanoparticles...what to do now Akash Reply it is a goid question and i want to know the answer as well Maciej characteristics of micro business Abigail for teaching engĺish at school how nano technology help us Anassong Do somebody tell me a best nano engineering book for beginners? s. Reply there is no specific books for beginners but there is book called principle of nanotechnology NANO what is fullerene does it is used to make bukky balls Devang Reply are you nano engineer ? s. fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball. Tarell what is the actual application of fullerenes nowadays? Damian That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes. Tarell what is the Synthesis, properties,and applications of carbon nano chemistry Abhijith Reply Mostly, they use nano carbon for electronics and for materials to be strengthened. Virgil is Bucky paper clear? CYNTHIA carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc NANO so some one know about replacing silicon atom with phosphorous in semiconductors device? s. Reply Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure. Harper Do you know which machine is used to that process? s. how to fabricate graphene ink ? SUYASH Reply for screen printed electrodes ? SUYASH What is lattice structure? s. Reply of graphene you mean? Ebrahim or in general Ebrahim in general s. Graphene has a hexagonal structure tahir On having this app for quite a bit time, Haven't realised there's a chat room in it. Cied what is biological synthesis of nanoparticles Sanket Reply how did you get the value of 2000N.What calculations are needed to arrive at it Smarajit Reply Privacy Information Security Software Version 1.1a Good A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place. Kimberly Reply Jeannette has$5 and \$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
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