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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

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
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The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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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
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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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