8.2
Writing Linear Equations
Lesson Objectives
Vocabulary
• Express the relationship between two quantities as a
linear equation.
linear equation
independent variable
dependent variable
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• Use a table or graph to represent a linear equation.
arn Write a linear equation to represent a given situation.
a)
Caleb is x years old. His sister is 10 years older than he is. If his sister is
y years old, write an equation that relates their two ages.
x
Caleb’s age
Sister’s age
x
10
From the model, an expression for the sister’s age is x 1 10.
To make an equation using this expression, notice that the problem says that the
sister’s age is another variable y.
So you can write:
y 5 x 1 10
The equation y 5 x 1 10 is called a linear equation.
In the equation, x is called the independent variable and y is called
the dependent variable because the value of y depends on the
value of x.
Writing y as an expression using x is called expressing y in terms
of x.
Continue on next page
Lesson 8.2 Writing Linear Equations
13
b)
A rhombus has sides of length r centimeters. If the perimeter of the rhombus
is P centimeters, express P in terms of r.
Expressing P in terms
of r means that r is the
r cm
independent variable, and
P is the dependent variable.
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r cm
r cm
r cm
r cm
From the model, the perimeter of the rhombus5 r 1 r 1 r 1 r
5 4r cm
Because the perimeter of the rhombus is P centimeters, you can write
P 5 4r.
Guided Practice
Complete.
1 Isaiah has h baseball cards. Miguel has 7 more baseball cards than Isaiah.
a) Write an expression for the number of baseball cards that Miguel has in
terms of h.
?
h
7
Miguel has
baseball cards.
b) If Miguel has k baseball cards, express k in terms of h.
k 5
?
c)
?
1
?
State the independent and dependent variables.
Independent variable:
14
? , dependent variable:
Chapter 8 Equations and Inequalities
?
Write an equation for each of the following. Then state the independent and
dependent variables for each equation.
2 Hannah took p minutes to jog around a park. Sofia took 12 minutes longer to
jog around the park. If Sofia took t minutes to jog around the park, express t
in terms of p.
3 A bouquet of roses costs $30. A bouquet of tulips costs m dollars less. If the
cost of one bouquet of tulips is n dollars, express n in terms of m.
4 Nathan has 7 boxes of marbles. Each box contains b marbles. If he has
c marbles altogether, express c in terms of b.
5 A motel charges Mr. Kim x dollars for his stay. Mr. Kim stayed at the motel for
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12 nights. If the rate per night for a room is y dollars, express y in terms of x.
arn Use tables and graphs to represent linear equations.
a)
The length of a rectangular picture frame is 5 inches longer than its width.
Write an equation to show how its width and length are related.
Let w represent the width of the picture frame, in inches.
Let , represent the length of the picture frame, in inches.
Since the length is 5 inches longer than the width,
, 5 w 1 5.
Many pairs of , and w values will make this equation true.
Width (w in.) Length (, in.)
1
2
3
4
5
15
15
15
15
15
6
7
8
9
10
The length is dependent on
the width.
The width (w) is the
independent variable, and the
length (,) is the dependent
variable.
Continue on next page
Lesson 8.2 Writing Linear Equations
15
The data on the previous page can be represented in a table, as shown
below. The first row of the table shows values of the independent variable.
The second row shows values of the dependent variable.
Width (w inches)
1
2
3
4
5
Length (, inches)
6
7
8
9
10
Use the data in the table to plot the ordered pairs (1, 6), (2, 7), (3, 8), (4, 9),
and (5, 10) in a coordinate plane. Connect the points to draw a line.
The horizontal axis shows the width of the picture frame.
The vertical axis shows the length of the picture frame.
Dimensions of a Picture Frame
,
11
10
9
Use the horizontal axis for
Length (in.)
8
the independent variable
7
and the vertical axis for
the dependent variable.
6
5
4
3
2
1
0
1
2
3
4
5
6
w
Width (in.)
All linear equations have graphs that are lines. The graph of a linear equation
contains all the ordered pairs that make the equation true.
For example, the point (2.5, 7.5) is on the graph of the equation , 5 w 1 5.
You can see that these values make the equation true:
, 5 w 1 5
7.5 5 2.5 1 5
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The ordered pair (2.5, 7.5) also makes sense in this situation because you could
have a picture frame that is 2.5 inches wide and 7.5 inches long.
Chapter 8 Equations and Inequalities
b)
Each can of paint contains 5 gallons of paint. Write an equation to show the
relationship between the number of cans and the volume of paint.
Let v represent the volume of paint,
and c represent the number of cans of paint.
Since each can contains 5 gallons of paint,
v 5 5c.
Cans (c )
Volume of paint in gallons (v )
The volume of paint depends on the
number of cans of paint. The number
of cans c is the independent variable,
and the volume of paint v is the
dependent variable.
Continue on next page
Lesson 8.2 Writing Linear Equations
17
The data on the previous page can be represented in a table, as shown below.
The first row of the table shows values of the independent variable,
and the second row shows values of the dependent variable.
Number of Cans (c )
1
2
3
4
5
Volume of Paint (v gallons)
5
10
15
20
25
You can write the data in the table as the ordered pairs (1, 5), (2, 10), (3, 15),
(4, 20), and (5, 25). These ordered pairs can be plotted in a coordinate plane,
and you can look for a pattern. You can connect the points to draw a line.
The horizontal axis shows the number of cans of paint.
The vertical axis shows the volume of paint.
Volume of Paint
v
Use the horizontal axis for
the independent variable
and the vertical axis for the
Volume of Paint (gal)
30
dependent variable.
25
20
15
10
5
0
1
2
3
4
5
6
c
Number of Cans of Paint
The graph of the equation v 5 5c contains all the ordered pairs that make
the equation true.
For example, the point (4.5, 22.5) is on the graph of the equation v = 5c.
You can see that these values make the equation true:
v 5 5c
22.5 5 5 3 4.5
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The ordered pair (4.5, 22.5) and other values of ordered pairs make sense in
this situation.
It is possible to have 4.5 cans of paint that contain 22.5 gallons of paint.
It is also possible to have 2
Chapter 8 Equations and Inequalities
1
2
cans of paint that contain 11 gallons of paint.
3
3
Guided Practice
Copy and complete the table. Then use the table to answer the questions.
6 The width of a rectangular tank is 2 meters less than its length.
a) If the length is p meters and the width is q meters, write an equation
relating p and q.
Length (p meters)
3
4
5
6
7
8
Width (q meters)
1
?
?
4
?
?
b) Use the data from a) to plot the points in a coordinate plane.
Connect the points with a line.
c)
The point (5.5, 3.5) is on the line you drew in b). Does this point make
sense in the situation?
Copy and complete each table. Then express the relationship between the
two variables as an equation.
7 Paul and Lee went to the library to borrow some books. Paul borrowed
6 more books than Lee.
Number of Books Borrowed by Lee (x)
1
2
3
4
5
Number of Books Borrowed by Paul (y)
7
?
9
?
?
8 At a crafts store, Zoey bought some boxes of red beads and some boxes of
blue beads. The number of boxes of red beads was 4 times the number of
boxes of blue beads.
Number of Boxes of Blue Beads (b)
2
3
4
5
6
Number of Boxes of Red Beads (r)
8
?
?
?
24
Use the data in the table to plot points in a coordinate plane.
Connect the points to form a line. Then write an equation to show the
relationship between the variables.
9
Time Taken (t hours)
1
2
3
4
5
6
Distance Traveled (d miles)
50
100
150
200
250
300
Lesson 8.2 Writing Linear Equations
19
Practice 8.2
Solve.
1 Joshua is w years old. His brother is 3 years older than he is.
a) If his brother is x years old, express x in terms of w.
b) State the independent and dependent variables in the equation.
2 Rita has b markers. Sandy has 11 fewer markers than she has.
a) If Sandy has h markers, express h in terms of b.
b) State the independent and dependent variables in the equation.
3 A small box of cereal weighs k grams. A jumbo box of cereal weighs 5 times
as much.
a) If the weight of the jumbo box of cereal is m grams, express m in terms
of k.
b) State the independent and dependent variables in the equation.
m grams
4 The area of Hank’s farm is n acres. The area of Hank’s farm is 8 times as large
as the area of Stan’s farm.
a) If s represents the area of Stan’s farm, express s in terms of n.
b) State the independent and dependent variables in the equation.
5 Ethan scored x points in a game. His younger sister scored 8 points when she
played the same game. Their combined score was y points.
a) Write an equation relating x and y.
b) Copy and complete the table to show the relationship between x and y.
Ethan’s Scores (x points)
Combined Scores ( y points)
20
Chapter 8 Equations and Inequalities
10
11
12
13
14
15
?
?
?
?
?
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k grams