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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

how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
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Sherica
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Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
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Kristine 2*2*2=8
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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J, combine like terms 7x-4y
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types of nano material
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I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
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what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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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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