SECTION 4.6
4.6
A Solutions and Graphs
B Special Cases
■
Graphing Equations in Two Variables
327
Graphing Equations in Two Variables
SU G G E S T I O N S
FOR SUCCESS
“That was then, this is now.” A person who studies mathematics in that way
fails to see how everything is tied together.
In Section 4.4, you learned how to find solutions of equations in two variables.
In this section, we will discuss how to draw a “picture” of those solutions. These
are not different and separate topics. In fact, they are the same topic.
Each day, you should cycle back and review previous material. When you
do, ask yourself how that material relates to what you are doing now. When
you do this, you will see how “that was then” is connected to “this is now.”
When you understand how topics flow together, you are on your way to
knowledge and success.
www
A Solutions and Graphs
LOPING
ONCEPT
HE C
DEVE
T
Plotting Points
As we saw in Section 4.5, an equation in two variables, such as x y 5, has
infinitely many ordered pair solutions. We know how to find as many of those
solutions as we want to. The following table lists eight of the solutions of
x y 5:
y
x
y
(x, y)
4
9
(4,
9)
2
7
(2,
7)
0
5
(0,
5)
1
4
(1,
4)
3
2
(3,
2)
5
0
(5,
0)
6
1
(6, 1)
9
4
(9, 4)
10
8
6
4
2
x
–10–8 –6 –4 –2
2 4
8 10
–4
–6
–8
–10
The figure beside the table shows these eight ordered pairs plotted in a rectangular coordinate system.
Do you notice that the eight points appear to be lined up? We could draw a
straight line that would contain every one of the points. In fact, if we were to
add 10 or 100 or 1,000 more solutions to the table and then plot those points,
they would continue to be points of that straight line.
Let’s be clear on the language that mathematicians like to use. The graph of
the equation x y 5 is a “picture” of all the solutions of that equation.
Because the points are all points of a line, we often refer to the graph as “the
line x y 5.”
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328
CHAPTER 4
■
Equations
Here is a summary of the procedure for graphing an equation whose graph is
a straight line.
L EARNING T IP
Only two points are needed
to draw a straight line, but
we recommend three. If the
three points appear to form a
triangle, you will know that
you have made an error in at
least one of them.
Graphing an Equation Whose Graph is a Straight Line
1. By selecting values for one variable and determining the corresponding
values of the other variable, find three solutions of the equation.
2. Plot these three ordered pairs.
3. Draw a line through the three points.
Example 1
Your Turn 1
Complete the following solutions of y x 4 and use
the solutions to draw the line that is the graph of the
equation.
(9,
) (4,
) (4,
)
SOLUTION
For x 9
For x 4
For x 4
y945
y440
y 4 4 8
(9, 5)
(4, 0)
(4, 8)
Complete the following solutions of
y x 3 and use the solutions to
draw the line that is the graph of
the equation.
(5,
(1,
(2,
)
)
)
y
y
10
8
6
4
2
y=x+3
(9, 5)
(4, 0)
x
–10–8 –6 –4 –2
–4
(–4, –8)
–8
–10
4 6 8 10
6
5
4
3
2
1
(2, 5)
(1, 4)
x
–6–5–4–3–2 –1
–2
(–5, –2) –3
–4
–5
–6
1 2 3 4 5 6
Answer: (5, 2), (1, 4), (2, 5)
In Example 1, we gave x-values to you in order to show how the graphs can be
drawn. When you study algebra, you will usually be asked to select your own
values. Two values that are often easy to use are x 0 and y 0.
Example 2
Draw the line 2x y 6.
Your Turn 2
Draw the line x 2y 4. Use
x 0, y 0, and choose another
value of y.
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