Transcript Graphs of Linear Equations in 2 Variables

Graphs; Equations of Lines;
Functions; Variation
Copyright © Cengage Learning. All rights reserved.
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Section
3.2
Graphs of Linear Equations
in Two Variables
Copyright © Cengage Learning. All rights reserved.
Objectives
11. Determine whether an ordered pair satisfies an
equation in two variables.
22. Construct a table of values given an equation.
33. Graph a linear equation in two variables by
constructing a table of values.
44. Graph a linear equation in two variables using
the intercept method.
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Objectives
5. Graph a horizontal line and a vertical line given
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an equation.
66. Write a linear equation in two variables from
given information, graph the equation, and
interpret the graphed data.
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1. Determine whether an ordered pair
satisfies an equation in two variables
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Equations in 2 Variables
The equation x + 2y = 5
• contains the two variables x and y
• solutions of such equations are ordered pairs of
numbers (x = 1, y = 2)
• For example, the ordered pair (1, 2) is a solution,
because the equation is satisfied when x = 1 and y = 2.
x + 2y = 5
1 + 2(2) = 5
1+4=5
5=5
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Example
Is the pair (–2, 4) a solution of y = 3x + 9?
Solution:
We substitute –2 for x and 4 for y and determine whether
the resulting equation is true.
y = 3x + 9
4 ≟ 3(–2) + 9
4 ≟ –6 + 9
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Example – Solution
cont’d
4=3
Since the equation 4 = 3 is false, the pair (–2, 4) is not a
solution.
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2.
Construct a table of values
given an equation
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Table of Solutions to an Equation
To find solutions of equations in x and y, we can pick
numbers at random, substitute them for x, and find the
corresponding values of y.
For example, to find some ordered pairs that satisfy
y = 5 – x, we can let x = 1(called the input value),
substitute 1 for x, and solve for y (called the output value).
y=5–x
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Table of Solutions to an Equation
y=5–1
y=4
The ordered pair (1, 4) is a solution. As we find solutions,
we will list them in a table of values like Table (1) below.
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Table of Solutions to an Equation
If x = 2, we have
y=5–x
y=5–2
y=3
This is the original equation.
Substitute the input value of 2 for x.
The output is 3.
A second solution is (2, 3). We list it in Table (2) below.
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Table of Solutions to an Equation
If x = 5, we have
y=5–x
y=5–5
y=0
This is the original equation.
Substitute the input value of 5 for x.
The output is 0.
A third solution is (5, 0). We list it in Table (3) below.
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Table of Solutions to an Equation
If x = –1, we have
y=5–x
y = 5 – (–1)
y=6
This is the original equation.
Substitute the input value of –1 for x.
The output is 6.
A fourth solution is (–1, 6). We list it in Table (4) below.
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Table of Solutions to an Equation
If x = 6, we have
y=5–x
y=5–6
y = –1
This is the original equation.
Substitute the input value of 6 for x.
The output is –1.
A fifth solution is (6, –1). We list it in Table (5) below.
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Table of Solutions to an Equation
Since we can choose any real number for x, and since any
choice of x will give a corresponding value of y, we can see
that the equation y = 5 – x has infinitely many solutions.
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3. Graph a linear equation in two variables
by constructing a table of values
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Graphing a Linear Equation in 2 Variables
A linear equation is any equation that can be written in the
form Ax + By = C, where A, B, and C are real numbers and
A and B are not both 0.
To graph the equation y = 5 – x, we plot the ordered pairs
listed in the table on a rectangular coordinate system, as in
Figure 3-10.
Figure 3-10
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Graphing a Linear Equation in 2 Variables
From the figure, we can see that the five points lie on a
line. We draw a line through the points. The arrowheads on
the line show that the graph continues forever in both
directions.
Since the graph of any solution of y = 5 – x will lie on this
line, the line is a picture of all of the solutions of the
equation. The line is said to be the graph of the equation.
Any equation, such as y = 5 – x, whose graph is a line is
called a linear equation in two variables. Any point on the
line has coordinates that satisfy the equation, and the
graph of any pair (x, y) that satisfies the equation is a point
on the line.
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Graphing a Linear Equation in 2 Variables
Since we usually will choose a number for x first and then
find the corresponding value of y, the value of y depends
on x. For this reason, we call y the dependent variable
and x the independent variable.
The value of the independent variable is the input value,
and the value of the dependent variable is the output value.
Although only two points are needed to graph a linear
equation, we often plot a third point as a check. If the three
points do not lie on a line, at least one of them is in error.
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Graphing a Linear Equation in 2 Variables
Graphing Linear Equations in Two Variables
1. Find two ordered pairs (x, y) that satisfy the equation by
choosing arbitrary input values for x and solving for the
corresponding output values of y. A third point provides
a check.
2. Plot each resulting pair (x, y) on a rectangular
coordinate system. If they do not lie on a line, check
your calculations.
3. Draw the line passing through the points.
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Example
Graph by constructing a table of values and plotting points:
y = 3x – 4.
Solution:
We find three ordered pairs that satisfy the equation.
If x = 1
y = 3x – 4
y = 3(1) – 4
y=1
If x = 2
y = 3x – 4
y = 3(2) – 4
y=2
If x = 3
y = 3x – 4
y = 3(3) – 4
y=5
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Example – Solution
cont’d
We enter the results in a table of values, plot the points,
and draw a line through the points. The graph appears in
Figure 3-11.
Figure 3-11
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4. Graph a linear equation in two
variables using the intercept method
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Graphing Linear Equation Using the
Intercept Method
The points where a line intersects the x- and y-axes are
called intercepts of the line.
x- and y-Intercepts
The x-intercept of a line is a
point (a, 0) where the line intersects
the x-axis. (See Figure 3-14.)
To find a, substitute 0 for y in
the equation of the line
and solve for x.
Figure 3-14
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Graphing Linear Equation Using the
Intercept Method
A y-intercept of a line is a point (0, b) where the line
intersects the y-axis. To find b, substitute 0 for x in the
equation of the line and solve for y.
Plotting the x- and y-intercepts and drawing a line through
them is called the intercept method of graphing a line.
This method is useful for graphing equations written in
general form.
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Graphing Linear Equation Using the
Intercept Method
General Form of the Equation of a Line
If A, B, and C are real numbers and A and B are not both 0,
then the equation
Ax + By = C
is called the general form of the equation of a line.
Comment
Whenever possible, we will write the general form
Ax + By = C so that A, B, and C are integers and A  0.
We also will make A, B, and C as small as possible.
For example, the equation 6x + 12y = 24 can be written as
x + 2y = 4 by dividing both sides by 6.
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Example
Graph by using the intercept method: 3x + 2y = 6.
Solution:
To find the y-intercept, we let x = 0 and solve for y.
3x + 2y = 6
3(0) + 2y = 6
Substitute 0 for x.
2y = 6
Simplify.
y=3
Divide both sides by 2.
The y-intercept is the point with coordinates (0, 3).
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Example – Solution
cont’d
To find the x-intercept, we let y = 0 and solve for x.
3x + 2y = 6
Substitute 0 for y.
3x + 2(0) = 6
Simplify.
3x = 6
Divide both sides by 3.
x=2
The x-intercept is the point with coordinates (2, 0).
As a check, we plot one more point. If x = 4, then
3x + 2y = 6
Substitute 4 for x.
3(4) + 2y = 6
Simplify.
12 + 2y = 6
Subtract 12 from both sides.
2y = –6
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Example – Solution
cont’d
y = –3
The point (4, –3) is on the graph. We plot these three
points and join them with a line.
The graph of 3x + 2y = 6 is
shown in Figure 3-15.
Figure 3-15
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5. Graph a horizontal line and a
vertical line given an equation
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Horizontal and Vertical Lines
Equations such as y = 3 and x = –2 are linear equations,
because they can be written in the general form
Ax + By = C.
y=3
x = –2
is equivalent to
is equivalent to
0x + 1y = 3
1x + 0y = –2
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Example
Graph:
a. y = 3
b. x = –2.
Solution:
a. We can write the equation y = 3 in general form as
0x + y = 3.
Since the coefficient of x is 0, the numbers chosen for x
have no effect on y. The value of y is always 3.
For example, if we substitute –3 for x, we get
0x + y = 3
0(–3) + y = 3
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Example – Solution
cont’d
0+y=3
y=3
The table in Figure 3-16(a) gives several pairs that
satisfy the equation y = 3.
Figure 3-16(a)
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Example – Solution
cont’d
After plotting these pairs and joining them with a line,
we see that the graph of y = 3 is a horizontal line that
intersects the y-axis at 3.
The y-intercept is (0, 3). There is no x-intercept.
b. We can write x = –2 in general form as x + 0y = –2.
Since the coefficient of y is 0, the values of y have no
effect on x. The value of x is always –2.
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Example – Solution
cont’d
A table of values and the graph are shown in
Figure 3-16(b).
Figure 3-16(b)
The graph of x = –2 is a vertical line that intersects the
x-axis at –2.
The x-intercept is (–2, 0). There is no y-intercept.
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Graphing Linear Equation Using the
Intercept Method
Equations of Horizontal and Vertical Lines
Suppose a and b are real numbers.
The equation y = b represents a horizontal line that
intersects the y-axis at (0, b). If b = 0, the line is the x-axis.
The equation x = a represents a vertical line that intersects
the x-axis at (a, 0). If a = 0, the line is the y-axis.
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6.
Write a linear equation in two variables from
given information, graph the equation, and
interpret the graphed data
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Linear Equation in 2 Variables
We have solved applications using one variable. In the next
example, we will write an equation containing two variables
to describe an application and then graph the equation.
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Example – Birthday Parties
A restaurant offers a party package that includes food,
drinks, cake, and party favors for a cost of $25 plus $3 per
child. Write a linear equation that will give the cost for a
party of any size. Graph the equation and determine the
meaning of the y-intercept in the context of this problem.
Solution:
Cost of party: c
Number of children: n
c = 25 + 3n
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Example – Solution
cont’d
c = 25 + 3n
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Example – Solution
If n = 0
c = 25 + 3(0)
c = 25
If n = 5
c = 25 + 3(5)
c = 25 + 15
c = 40
cont’d
If n = 10
c = 25 + 3(10)
c = 25 + 30
c = 55
The results are recorded in the table.
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Example – Solution
cont’d
Next, we graph the points in Figure 3-17 and draw a line
through them.
We don’t draw an arrowhead
on the left, because it doesn’t
make sense to have a negative
number of children attend a party.
From the graph, we can determine
the y-intercept is (0, 25).
The $25 represents the setup
cost for a party with no attendees.
Figure 3-17
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Example – Solution
cont’d
Comment
The scale for the cost (y-axis) is 5 units and the scale for
the number attending (x-axis) is 1.
Since the scales on the x- and y-axes are not the same,
you must label them!
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