Download 8.2 Introduction to Graphing Linear Equations So far we have dealt

Math 40
Prealgebra
Section 8.2 – Introduction to Graphing Linear Equations
8.2 Introduction to Graphing Linear Equations
So far we have dealt with linear equations in one variable such as 3x  4  11 . It has only one solution, x  5 ,
since 3  5   4  11 . This is always the case for linear equations in one variable.
In this section, we consider linear equations in two variables such as y  x  2 . A solution to this type of equation
is an ordered pair  x, y  that satisfies the equation. For example  4, 6  is a solution since replacing x with 4 and
y with 6 gives us a true statement:
y  x2
?
6  42
66
TRUE
1 1
, 5  are some solutions to
 2 2
name a few. We can verify this by substituting the x-coordinate and y-coordinate of each ordered pair into
y  x  2 and obtaining a true statement.
Can we find more ordered pairs that satisfy y  x  2 ? Yes!
y  x2
 7,  5 ,  2, 0  ,  3
y  x2
y  x2
?
?
5   7  2
0  22
5   5
00
TRUE
T RUE
1 ? 1
5 3 2
2
2
1
1
5 5
TRUE
2
2
It is important to note that one can find infinitely many ordered pairs to satisfy a linear equation in two
variables. We will discuss this in more detail throughout this section. In the meanwhile, let’s practice our
understanding of what it means to be a solution to a linear equation in two variables.
Example 1: Determine if the following ordered pairs are solutions of the equation y  2 x  8 .
a)
Solution:
 8, 11
b)
 1,  6 
c)
 2,  14 
d)  3,  13 
a) To determine if  8, 11 is a solution of y  2 x  8 , replace x with 8 and y with 11 :
y  2 x  8
?
11   2  8   8
?
11  16  8
?
11  8
FALSE
Since the resulting statement is false,  8, 11 is not a solution of y  2 x  8 .
1
2015 Campeau
Math 40
Prealgebra
Section 8.2 – Introduction to Graphing Linear Equations
b) To determine if  1,  6  is a solution of y  2 x  8 , replace x with 1 and y with 6 :
y  2 x  8
?
6   2  1  8
?
6  2  8
?
6   6
TRUE
Since the resulting statement is true,  1,  6  is a solution of y  2 x  8 .
c) To determine if  2,  14  is a solution of y  2 x  8 , replace x with 2 and y with 14 :
y  2 x  8
?
14   2  2   8
?
14   4  8
?
14   12
FALSE
Since the resulting statement is false,  2,  14  is not a solution of y  2 x  8 .
d) To determine if  3,  13 is a solution of y  2 x  8 , replace x with 3 and y with 13 :
y  2 x  8
?
13   2  3   8
?
13   6  8
?
13   14
FALSE
Since the resulting statement is false,  3,  13  is not a solution of y  2 x  8 .
You Try It 1: Determine if the following ordered pairs are solutions of the equation y  3x  4 .
a)
 6, 20 
b)
 4,  16 
c)
 4,  8 
d)
 6,  14 
2
2015 Campeau
Math 40
Prealgebra
Section 8.2 – Introduction to Graphing Linear Equations
Example 2: Find the missing coordinate so that the ordered pair will satisfy the given linear equation.
a) y  5 x  7;
Solution:
 6, ? 
b) y  4 x  9;
 ?, 1
 1
c) 2 x  6 y  12;  ?, 
 3
a) In  6, ?  the x-coordinate is 6 and the y-coordinate is unknown.
Replace x with 6 in y  5x  7 and solve for y:
y  5  6   7
y  30  7
The ordered pair  6,  23 satisfies y  5x  7 .
y  23
b) In  ?, 1 the x-coordinate is unknown and the y-coordinate is 1 .
Replace y with 1 in y  4x  9 and solve for x:
y  4 x  9
1  4 x  9
10  4 x
10
x
4
5
 x
2
1
2  x
2
 1 
The ordered pair  2 , 1 satisfies y  4x  9 .
 2 
1
 1
c) In  ?,  the x-coordinate is unknown and the y-coordinate is .
3
 3
Replace y with
1
in 2 x  6 y  12 and solve for x:
3
2 x  6 y  12
1
2 x  6    12
3
2 x  2  12
2 x  14
 1
The ordered pair  7,  satisfies 2 x  6 y  12 .
 3
x7
3
2015 Campeau
Math 40
Prealgebra
Section 8.2 – Introduction to Graphing Linear Equations
You Try It 2: Find the missing coordinate so that the ordered pair will satisfy the given linear equation.
 1 
a) y  8 x  5;   , ? 
 4 
b) y   x  3;
 ?,  1
c) 3 x  4 y  12;
 2, ? 
Now that we have had some practice with what it means to be a solution (and what it means to not be a
solution) of a linear equation in two variables, we focus our attention on finding all solutions of a linear
equation in two variables.
Let’s consider the equation y  3x  4 . Since this equation is solved for y, y depends on the value of x that is
substituted into the equation.
We can quickly calculate ordered pairs that are solutions of y  3x  4 by choosing many values for x.
To make it easier on ourselves, we choose integer values of x and calculate the corresponding y-value for each.
x  4 : y  3  4   4  12  4  8
y  3x  4
x   3 : y  3   3   4   9  4  5
x
y
x  2 : y  3  2   4   6  4  2
4
8
x  1 : y  3  1  4  3  4  1
3
5
x  0 : y  3 0  4  0  4  4
2
2
x  1 : y  3 1  4  3  4  7
1
1
0
4
1
7
2
10
x  2 : y  3  2   4  6  4  10
(x, y)
 4,  8 
 3,  5 
 2,  2 
 1, 1
 0, 4 
 1, 7 
 2, 10 
We organize the results in a table so we can easily refer to it while graphing the ordered pairs.
4
2015 Campeau
Math 40
Prealgebra
Section 8.2 – Introduction to Graphing Linear Equations
We plot the ordered pairs below:
10
y
8
6
4
2
x
10 8
6
4
2
0
2
4
6
8
10
2
4
6
8
10
This is only a plot of seven solutions of y  3x  4 . We can find more solutions by choosing non-integer values
of x and calculating the corresponding y-value for each. To make it easier on ourselves, we will choose xvalues in between the integer values we found earlier.
x  4.5 : y  3  4.5   4  13.5  4  9.5
y  3x  4
x  3.5 : y  3  3.5   4  10.5  4  6.5
x
y
x  2.5 : y  3  2.5   4  7.5  4  3.5
4.5
9.5
x  1.5 : y  3  1.5   4  4.5  4  0.5
3.5
6.5
x  0.5 : y  3  0.5   4  1.5  4  2.5
2.5
3.5
x  0.5 : y  3  0.5   4  1.5  4  5.5
1.5
0.5
x  1.5 : y  3 1.5   4  4.5  4  8.5
0.5
2.5
0.5
5.5
1.5
8.5
(x, y)
 4.5,  9.5 
 3.5,  6.5 
 2.5,  3.5 
 1.5,  0.5 
 0.5, 2.5 
 0.5, 5.5 
 1.5, 8.5 
Again, we organize the results in a table so we can easily refer to it while graphing the ordered pairs.
5
2015 Campeau
Math 40
Prealgebra
Section 8.2 – Introduction to Graphing Linear Equations
Plotting the six ordered pairs we just found, along with the seven ordered pairs we found earlier, gives us the
following:
10
y
8
6
4
2
x
10 8
6
4
2
0
2
4
6
8
10
2
4
6
8
10
We can continue to choose more and more x-values and add more and more points to the graph. Eventually we
will have so many points plotted that the graph of the solutions resembles a straight line.
10
y
8
6
4
2
x
10 8
6
4
2
0
2
4
6
8
10
2
4
6
8
10
By drawing a line through the ordered pairs we plotted, we are representing the infinitely many ordered pairs
that satisfy the equation y  3x  4 . The line represents the graph of the linear equation y  3x  4 .
Note: In order to get an accurate representation of the graph of a linear equation, it is very important
that you use a straight edge to connect the points you plot. A ruler, student id card, or sheet of
paper folded in half will serve us nicely for the problems in this section.
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2015 Campeau
Math 40
Prealgebra
Section 8.2 – Introduction to Graphing Linear Equations
You may be wondering if you need to choose as many x-values as we did in our graph of y  3x  4 .
Luckily for us, it has been determined that the minimum number of ordered pairs needed to graph a linear
equation is two ordered pairs. However, we recommend to find at least three ordered pairs. Finding at least
three ordered pairs is can be helpful; if you happen to make a minor error in your calculations, you will catch it
right away when you see that the ordered pairs don’t fall in a straight line.
You also may be wondering if you are allowed to choose any x-values you want.
Pay attention to the directions of the problem. Sometimes, you are given a table of specific x-values for which
you must calculate the corresponding y-values and sometimes you are asked to construct your own table of
values.
Example 3: Graph the linear equation by completing the table of values.
y  3x  1
x
Solution:
(x, y)
y
2
Substitute x  2 into y  3x  1 and solve for y.
y  3   2   1  6  1  7
0
Record the result in the table. Plot the ordered pair.
3
Substitute x  0 into y  3x  1 and solve for y.
y  3  0   1  0  1  1 Record the result in the table. Plot the ordered pair.
Substitute x  3 into y  3x  1 and solve for y.
y  3  3   1  9  1  8 Record the result in the table. Plot the ordered pair.
10
y  3x  1
y
2
7
 2, 7 
0
1
 0, 1
3
8
6
(x, y)
x
y
8
4
2
x
10 8
6
 3,  8 
4
2
0
2
4
6
8
10
2
4
6
Use a straight edge to connect the plotted points.
8
10
Make sure to extend the line through both edges of the axes!
7
2015 Campeau
Math 40
Prealgebra
Section 8.2 – Introduction to Graphing Linear Equations
You Try It 3: Graph the linear equation by completing the table of values.
10
y  2x  4
x
y
(x, y)
y
3
8
6
0
4
6
2
x
10 8
6
4
2
0
2
4
6
8
10
2
4
6
8
10
Example 4: Graph y   x  7 by constructing a table of values.
Solution:
Here we have the freedom to choose any values of x we desire. We recommend a minimum of
three x-values. The following is a possible solution.
It is important to note that choosing different x-values will give us different ordered pairs,
however, the final graph will look the same.
Substitute x  5 into y   x  7 and solve for y: y    5   7  5  7  2
Substitute x  0 into y   x  7 and solve for y: y    0   7  0  7  7
Substitute x  3 into y   x  7 and solve for y:
y    3  7  3  7  10
10
y  x  7
x
y
8
(x, y)
5
2
 5,  2 
0
7
 0,  7 
3
10
 3,  10 
y
6
4
2
x
10 8
6
4
2
0
2
4
6
8
10
2
4
Use a straight edge to connect the plotted points.
6
8
Make sure to extend the line.
10
8
2015 Campeau
Math 40
Prealgebra
Section 8.2 – Introduction to Graphing Linear Equations
You Try It 4: Graph y  x  5 by constructing a table of values.
10
y
8
6
4
2
x
10 8
6
4
2
0
2
4
6
8
10
2
4
6
8
10
3
Example 5: Graph y   x  2 by constructing a table of values.
4
Solution:
Here we have the freedom to choose any values of x we desire. We recommend a minimum of
three x-values.
Notice that the coefficient of x is a fraction. If we want to ensure we only obtain integer values of
y in our solution, we choose values of x that are multiples of the denominator.
It is still ok if you decide to choose values of x that are not multiples of the denominator, however,
you will obtain fractional or decimal values of y and you will have to be more careful when
plotting the ordered pairs you obtain.
The following is a possible solution.
3
3
Substitute x  4 into y   x  2 and solve for y: y    4   2  3  2  5
4
4
3
3
Substitute x  4 into y   x  2 and solve for y: y    4   2  3  2  1
4
4
3
Substitute x  8 into y   x  2 and solve for y:
4
y
3
8  2  6  2  4
4
9
2015 Campeau
Math 40
Prealgebra
Section 8.2 – Introduction to Graphing Linear Equations
x
4
4
8
y
5
1
 4,  1
4
8
6
(x, y)
 4, 5 
y
10
3
y  x2
4
4
2
x
10 8
6
4
2
0
2
4
6
8
10
2
 8,  4 
4
6
8
Use a straight edge to connect the plotted points.
10
Make sure to extend the line.
You Try It 5: Graph y 
3
x  4 by constructing a table of values.
2
10
y
8
6
4
2
x
10 8
6
4
2
0
2
4
6
8
10
2
4
6
8
10
10
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