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Graph the equation: x + y = −2 .


This answer graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  The equation x plus y equals -2 is  shown. A line passes through the intercepts with coordinates 0, –2 and –2, 0.

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Graph the equation: x y = 6 .


The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 6, 0” and “ordered pair 0, -6”.

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In the previous example, the three points we found were easy to graph. But this is not always the case. Let’s see what happens in the equation 2 x + y = 3 . If y is 0 , what is the value of x ?

This figure shows an algebraic substitution. The first line is 2 x + y = 3. The second line is 2 x + 0, with 0 shown in red. The third line is 2 x = 3. The last line is x = 3 over 2.

The solution is the point ( 3 2 , 0 ) . This point has a fraction for the x -coordinate. While we could graph this point, it is hard to be precise graphing fractions. Remember in the example y = 1 2 x + 3 , we carefully chose values for x so as not to graph fractions at all. If we solve the equation 2 x + y = 3 for y , it will be easier to find three solutions to the equation.

2 x + y = 3
y = −2 x + 3

Now we can choose values for x that will give coordinates that are integers. The solutions for x = 0 , x = 1 , and x = −1 are shown.

y = −2 x + 3
x y ( x , y )
0 5 ( −1 , 5 )
1 3 ( 0 , 3 )
−1 1 ( 1 , 1 )
The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through three labeled points, “ordered pair -1, 5”, “ordered pair 0, 3”, and ordered pair 1, 1”. The line is labeled 2 x + y = 3.

Graph the equation 3 x + y = −1 .

Solution

Find three points that are solutions to the equation.

First, solve the equation for y .

3 x + y = −1 y = −3 x 1

We’ll let x be 0 , 1 , and −1 to find three points. The ordered pairs are shown in the table. Plot the points, check that they line up, and draw the line.

y = −3 x 1
x y ( x , y )
0 −1 ( 0 , −1 )
1 −4 ( 1 , −4 )
−1 2 ( −1 , 2 )
The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through three labeled points, “ordered pair -1, 2”, “ordered pair 0, -1”, and ordered pair 1, -4”. The line is labeled 3 x + y = -1.

If you can choose any three points to graph a line, how will you know if your graph matches the one shown in the answers in the book? If the points where the graphs cross the x - and y -axes are the same, the graphs match.

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Graph each equation: 2 x + y = 2 .


The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through three labeled points, “ordered pair -1, 2”, “ordered pair 0, -1”, and ordered pair 1, -4”. The line is labeled 3 x + y = -1.

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Graph each equation: 4 x + y = −3 .


The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A line passes through the points “ordered pair -1,  4” and “ordered pair 0, -3”.

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Graph vertical and horizontal lines

Can we graph an equation with only one variable? Just x and no y , or just y without an x ? How will we make a table of values to get the points to plot?

Let’s consider the equation x = −3 . The equation says that x is always equal to −3 , so its value does not depend on y . No matter what y is, the value of x is always −3 .

To make a table of solutions, we write −3 for all the x values. Then choose any values for y . Since x does not depend on y , you can chose any numbers you like. But to fit the size of our coordinate graph, we’ll use 1 , 2 , and 3 for the y -coordinates as shown in the table.

x = −3
x y ( x , y )
−3 1 ( −3 , 1 )
−3 2 ( −3 , 2 )
−3 3 ( −3 , 3 )

Then plot the points and connect them with a straight line. Notice in [link] that the graph is a vertical line    .

The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A vertical line passes through three labeled points, “ordered pair -3, 3”, “ordered pair -3, 2”, and ordered pair -3, 1”. The line is labeled x = -3.

Vertical line

A vertical line is the graph of an equation that can be written in the form x = a .

The line passes through the x -axis at ( a , 0 ) .

Graph the equation x = 2 . What type of line does it form?

Solution

The equation has only variable, x , and x is always equal to 2 . We make a table where x is always 2 and we put in any values for y .

x = 2
x y ( x , y )
2 1 ( 2 , 1 )
2 2 ( 2 , 2 )
2 3 ( 2 , 3 )

Plot the points and connect them as shown.

The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A vertical line passes through three labeled points, “ordered pair 2, 3”, “ordered pair 2, 2”, and ordered pair 2, 1”. The line is labeled x = 2.

The graph is a vertical line passing through the x -axis at 2 .

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Graph the equation: x = 5 .


The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A vertical line passes through the points “ordered pair 5,  0” and “ordered pair 5, 1”.

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Graph the equation: x = −2 .


The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12. A vertical line passes through the points “ordered pair 5,  0” and “ordered pair 5, 1”.

Got questions? Get instant answers now!

What if the equation has y but no x ? Let’s graph the equation y = 4 . This time the y -value is a constant, so in this equation y does not depend on x .

To make a table of solutions, write 4 for all the y values and then choose any values for x .

We’ll use 0 , 2 , and 4 for the x -values.

y = 4
x y ( x , y )
0 4 ( 0 , 4 )
2 4 ( 2 , 4 )
4 4 ( 4 , 4 )

Plot the points and connect them, as shown in [link] . This graph is a horizontal line    passing through the y -axis at 4 .

The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A horizontal  line passes through three labeled points, “ordered pair 0, 4”, “ordered pair 2, 4”, and ordered pair 4, 4”. The line is labeled y = 4.

Horizontal line

A horizontal line is the graph of an equation that can be written in the form y = b .

The line passes through the y -axis at ( 0 , b ) .

Questions & Answers

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Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
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Application of nanotechnology in medicine
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please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
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Professor
I think
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Nasa has use it in the 60's, copper as water purification in the moon travel.
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Rafiq
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Damian
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LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
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Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
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biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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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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