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By the end of this section, you will be able to:
  • Recognize the relation between the solutions of an equation and its graph
  • Graph a linear equation by plotting points
  • Graph vertical and horizontal lines

Before you get started, take this readiness quiz.

  1. Evaluate: 3 x + 2 when x = −1 .
    If you missed this problem, review Multiply and Divide Integers .
  2. Solve the formula: 5 x + 2 y = 20 for y .
    If you missed this problem, review Solve a Formula for a Specific Variable .
  3. Simplify: 3 8 ( −24 ) .
    If you missed this problem, review Multiply and Divide Fractions .

Recognize the relation between the solutions of an equation and its graph

In Use the Rectangular Coordinate System , we found a few solutions to the equation 3 x + 2 y = 6 . They are listed in the table below. So, the ordered pairs ( 0 , 3 ) , ( 2 , 0 ) , ( 1 , 3 2 ) , ( 4 , 3 ) , are some solutions to the equation 3 x + 2 y = 6 . We can plot these solutions in the rectangular coordinate system as shown on the graph at right.

...

Notice how the points line up perfectly? We connect the points with a straight line to get the graph of the equation 3 x + 2 y = 6 . Notice the arrows on the ends of each side of the line. These arrows indicate the line continues.

...

Every point on the line is a solution of the equation. Also, every solution of this equation is a point on this line. Points not on the line are not solutions!

Notice that the point whose coordinates are ( 2 , 6 ) is on the line shown in [link] . If you substitute x = 2 and y = 6 into the equation, you find that it is a solution to the equation.

...

So ( 4 , 1 ) is not a solution to the equation 3 x + 2 y = 6 . Therefore the point ( 4 , 1 ) is not on the line.

This is an example of the saying,” A picture is worth a thousand words.” The line shows you all the solutions to the equation. Every point on the line is a solution of the equation. And, every solution of this equation is on this line. This line is called the graph of the equation 3 x + 2 y = 6 .

Graph of a linear equation

The graph of a linear equation A x + B y = 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.

The graph of y = 2 x 3 is shown below.

...

For each ordered pair decide

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

  1. ( 0 , 3 )
  2. ( 3 , 3 )
  3. ( 2 , 3 )
  4. ( 1 , 5 )

Substitute the x - and y -values into the equation to check if the ordered pair is a solution to the equation.


...

Plot the points A: ( 0 , 3 ) B: ( 3 , 3 ) C: ( 2 , 3 ) and D: ( 1 , 5 ) .
The points ( 0 , 3 ) , ( 3 , 3 ) , and ( 1 , 5 ) are on the line y = 2 x 3 , and the point ( 2 , 3 ) is not on the line.

...

The points which are solutions to y = 2 x 3 are on the line, but the point which is not a solution is not on the line.

Got questions? Get instant answers now!

The graph of y = 3 x 1 is shown.

For each ordered pair, decide

  1. is the ordered pair a solution to the equation?
  2. is the point on the line?
...
  1. ( 0 , 1 )
  2. ( 2 , 2 )
  3. ( 3 , 1 )
  4. ( 1 , 4 )
  1. yes yes
  2. no no
  3. no no
  4. yes yes
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Graph a linear equation by plotting points

There are several methods that can be used to graph a linear equation. The method we used at the start of this section to graph is called plotting points, or the Point-Plotting Method .

Let’s graph the equation y = 2 x + 1 by plotting points.

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.
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Damian Reply
absolutely yes
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it is a goid question and i want to know the answer as well
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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
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On having this app for quite a bit time, Haven't realised there's a chat room in it.
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Practice Key Terms 2

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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