Applications of Linear Equations Notes
Solving an Applied Problem
Step 1 Read the problem, several times if necessary, until you understand what is given
and what is given and what is to be found.
Step 2 Assign a variable to represent the unknown value, using diagrams or tables as
needed. Write down what the variable represents. Express any other unknown
values in terms of the variable.
Step 3 Write an equation using the variable expression(s).
Step 4 Solve the equation.
Step 5 State the answer. Does it seem reasonable?
Step 6 Check the answer in the words of the original problem.
Note: The third step in solving an applied problem is often the hardest. To translate the
problem into an equation, write the given phrases as mathematical expressions.
Replace any words that mean equal or same with an = sign. Other forms of the
verb “to be,” such as is, are, was, and were, also translate as an = sign. The =
sign leads to an equation to be solved.
Type 1: Solve Problems Involving Unknown Numbers.
Example 1-1. Finding the Value of an Unknown Number
The product of 4, and a number decreased by 7, is 100. What is the number?
Step 1 Read the problem carefully. We are asked to find a number.
Step 2 Assign a variable to represent the unknown quantity. In this problem, we are
asked to find a number, so we write
Let x = number
The are no other unknown quantities to find.
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-1-
Applications of Linear Equations Notes
-2-
Step 3 Write an equation.
the
product
of 4,
4
and
Step 4 Solve the equation.
a
decreased
number
by
(x
-
4x  7  100
4x  28  100
4 x  28  100
 28
4x  128
4 x 128

4
4
x  32
 28
7,
7)
is
=
100
100
Distributive property
Add 28
Combine terms
Divide by 4
Step 5 State the answer. The number is 32
Step 6 Check. When 32 is decreased by 7, we get 32-7=25. If 4 is multiplied by 25, we
get 100, as required. The answer 32, is correct.
Example 1-2. Finding the Value of an Unknown Number
The product of 8, and a number increased by 6, is 104. What is the number?
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Applications of Linear Equations Notes
Example 1-3. Finding the Value of an Unknown Number
The product of 5, and 3 more than twice a number, is 85. What is the number?
Example 1-4. Finding the Value of an Unknown Number
Two less than three times a number is equal to 14 more than five times the number.
What is the number?
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-3-
Applications of Linear Equations Notes
Example 1-5. Finding the Value of an Unknown Number
Nine more than five times a number is equal to 3 less than seven times the number. What
is the number?
Example 1-6. Finding the Value of an Unknown Number
If 2 is subtracted from a number and this difference is tripled, the result is 6 more than the
number. Find the number.
Example 1-7. Finding the Value of an Unknown Number
If 3 is added to a number and this sum is doubled, the result is 2 more than the number.
Find the number?
Homework on Type 1:
Barron’s Book p.33 (11-17) All work must be shown on separate paper.
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-4-
Applications of Linear Equations Notes
-5-
Type 2: Problems Involving Sums of Quantities.
Example 2-1. Finding Numbers of Olympic Medals
In the 2002 Winter Olympics in Salt Lake City, the United States won 10 more medals
than Norway. The two countries won a total of 58 medals. How many medals
did each country win?
Step 1 Read the problem carefully. We are given information about the total number of
medals and asked to find the number each country won. The total number of
medals is 58.
Step 2 Assign a variable.
Norway’s Medals
x
United States won 10 more medals than
US Medals
Norway
x+10
Let x = the number of medals Norway won
Then x + 10 = the number of medals the US won.
You must write these statements.
Step 3 Write an equation.
The
total
58
is
=
Norway
Medals
x
plus
+
US
medals
(10+x)
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Applications of Linear Equations Notes
Step 4 Solve the equation.
-6-
58  x  (10  x )
58  x  x  10  2 x  10
58  2 x  10
 10
 10
48  2 x
48 2 x

2
2
24  x
Step 5 State the answer.
Norway’s Medals
Combine terms
Subtract 10
Combine terms
Divide by 2
x
x=24
x+10
24+10=34
United States won 10 more medals than
US Medals
Norway
Step 6 Check. Since the United States won 34 medals and Norway won 24, the total
number of medals was 34 + 24 = 58. Because 34 – 24 = 10, the United States won 10
more medals than Norway. This information agrees with what is given in the problem, so
the answer checks.
Problem-Solving Hint
Example 2-1 could also be solved by letting x represent the number of medals the
US own. Then x-10 would represent the number of medals Norway won. The
equation would be:
58  x  x  10
The solution of this equation is 34, which is the number of US medals. The number
of Norwegian medals would be 34-10 = 24. The answers are the same, whichever
approach is used.
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Applications of Linear Equations Notes
-7-
Example 2-2. Finding the Value of Three Unknown Numbers
Working with unknown quantities.
The instructions for a woodworking project call for three pieces of wood. The longest
piece must be twice the length of the middle-sized piece, and the shortest piece must be
10 in shorter than the middle-sized piece. Maria has a board 70 in long that she wishes to
use. How long must each piece be?
Step 1 Read the problem carefully. Three lengths must be found.
Step 2 Assign a variable. Since the middle-sized piece appears in both pairs of
comparisons, let x represent the length, in inches, of the middle-sized piece. We
have…
Let x represent the length, in inches, of the middle-sized piece.
Shortest Piece
10 in shorter than the middle-sized piece
Middle Piece
Longest Piece
x-10
x
twice the length of the middle-sized piece
2x
Step 3 Write an equation. (sometimes a picture is helpful)
Longest
2x
and
+
Middle
x
and
+
Shortest
(x-10)
is
=
70
70
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Applications of Linear Equations Notes
Step 4 Solve the equation.
-8-
2x  x  x  10  70
4x  10  70
4 x  10  70
 10
 10
4x  80
4 x 80

4
4
x  20
Combine terms
Add 10
Combine terms
Divide by 4
Step 5 State the answer.
Shortest Piece
10 in shorter than the middle-sized piece
Middle Piece
Longest Piece
twice the length of the middle-sized piece
x-10
20-10=10
x
20
2x
2(20)=40
Step 6 Check. 10+20+40 is 70 in. All conditions in the problem are satisfied.
Example 2-3. Problems Involving Sums of Quantities.
The number of drive-in movie screens has declined steadily in the United States since the
1960s. California and New York were two of the states with the most remaining drive-in
movie screens in 2001. California had 11 more screens than New York, and there were
107 screens total in the two states. How many drive-in movie screens remained in each
state?
________________________________________________________________________
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Applications of Linear Equations Notes
Example 2-4. Problems Involving Sums of Quantities.
The total number of Democrats and Republicans in the US House of Representatives
during the 108th session was 434. There were 24 more Republicans than Democrats.
How many members of each party were there?
Example 2-5. Problems Involving Sums of Quantities.
The Toyota Camry was the top-selling passenger car in the US in 2003, followed by the
Honda Accord. Honda Accord sales were 35 thousand less than Toyota Camry sales, and
833 thousand of these two cars were sold. How many of each make of car were sold?
________________________________________________________________________
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-9-
Applications of Linear Equations Notes
Example 2-6. Problems Involving Sums of Quantities.
A lawn trimmer uses a mixture of gasoline and oil. The mixture contains 16 oz of
gasoline for each ounce of oil. If the tank holds 68 oz of the mixture, how many
ounces of oil and how many ounces of gasoline does it require when it is full?
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- 10 -
Applications of Linear Equations Notes
Homework Type 2 You must show all work on separate paper.
Write an equation for each of the following and then solve the problem.
Homework 1:
1. If 4 is added to 3 times a number, the result is 7. Find the number.
2. If 2 is subtracted from four times a number, the result is 3 more than six times the
number. What is the number?
3. If -2 is multiplied by the difference between 4 and a number, the result is 24.
Find the number.
4. George and Al were opposing candidates in the school board election. George
received 21 more votes than Al, with 439 votes cast. How many votes did Al
receive?
5. A rope 116 inches long is cut into three pieces. The middle-sized piece is 10
inches shorter than twice the shortest piece. The longest piece is as long as the
shortest piece. What is the length of the shortest piece?
6. On a psychology test, the highest grade was 38 points more than the lowest grade.
The sum of the two grades was 142. Find the lowest grade.
Homework 2:
7. Mount McKinley in Alaska is 5910 feet higher than Mount Rainier in
Washington. Together, their heights total 34, 730 feet. How high is each
mountain?
8. Penny is make punch for a party. The recipe requires twice as much orange juice
as cranberry juice and 8 times as much ginger ale as cranberry juice. If she plans
to make 176 ounces of punch, how much of each ingredient should she use?
9. Pablo, Frank, and Mark swim at a public pool each day for exercise. One day
Pablo swam five more than three times as many laps as Mark, and Frank swam
four times as many laps as Mark. If the men swam 29 laps altogether, how many
laps did each one swim?
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- 11 -
Applications of Linear Equations Notes
10. Linda wishes to build a rectangular dog pen using 52 feet of fence and the back of
her house, which is 36 feet long to enclose the pen. How wide will the dog pen be
if the pen is 36 feet long?
Solutions
1. 1
2. −
3. 16
4. 209 votes
5. 27 inches
6. 52
7. Mount McKinley: 20,320 ft
Mount Rainier: 14, 410 ft
8. 16 oz cranberry juice
32 oz of orange juice
128 oz of ginger ale
9. Mark 3, Pablo 14, Frank 12
10. 8 ft
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- 12 -
Applications of Linear Equations Notes
- 13 -
Type 3: Problems Involving Supplementary and
Complementary Angles
Before we solve these types of problems, you need to know several definitions.
Two angles are
when their sum Definition
called….
is…
Complementary Angles 90°
If two angles are complementary, then the
measures of the angles add up to 90°.
Complementary angles may have, but are not
required to have, a shared side.
Example:
Supplementary Angle
180°
If two angles are supplementary, then the
measures of the angles add up to 180°.
Supplementary angles may have, but are not
required to have, a shared side.
Hint:
 "C" of Complementary stands for "Corner"
(a Right Angle), and
 "S" of Supplementary stands for "Straight" (180 degrees is a straight line)
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Applications of Linear Equations Notes
Straight Angle:
A straight angle is 180°.
Problem Solving Hint:
If x represents the degree measure of an angle, then
90 − represents the degree measure of its complement
180 − represents the degree measure of its supplement
Vertical Angle
Vertical angles are two nonadjacent angles formed when two lines intersect. If two
angles are vertical angles, then they have equal measures (are congruent).
In this example, a° and b° are vertical angles.
The interesting thing here is that vertical angles are equal:
a° = b°
Example 3-1. Finding the Measure of an Angle
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- 14 -
Applications of Linear Equations Notes
- 15 -
Finding the measure of an angle whose supplement is 10° more than twice its
complement.
Step 1 Read the problem carefully. We are to find the measure of an angle, given
information about its complement and its supplement.
Step 2 Assign a variable.
Let…
x=
the degree measure of the angle
− =
18 −
The degree measure of its complement
=
The degree measure of its supplement
Step 3 Write an equation.
Supplement
180-x
is
=
10
10
more
than
+
twice
2
its
complement
(90-x)
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Applications of Linear Equations Notes
Step 4 Solve the equation.
- 16 -
180  x  10  2(90  x )
180  x  10  180  2 x
180  x  190  2 x
180  x  190  2 x
 2x
 2x
180  x  190
180  x  190
 180
 180
Distributive Property
Combine terms
Add 2x
Combine terms
Subtract 180
= 10
Step 5 State the answer.
x=
the degree measure of the angle
− =
18 −
The degree measure of its complement
=
The degree measure of its supplement
The measure of the angle is 10°
x =10
−
18 −
= 90 − 10 = 80
= 180 − 80 = 100
Step 6 Check. The complement of 10° is 80° and the supplement of 10° is 170°. Also,
170° is equal to 10° more than twice 80°. Therefore, the answer is correct.
Example 3-2. Finding the Measure of an Angle
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Applications of Linear Equations Notes
Example 3-3. Finding the Measure of an Angle
The complement of an angle is 46°. What is the measure of the angle?
Example 3-4. Finding the Measure of an Angle
Twice the complement of angle A is 40° less than the supplement of angle A. Find the
measure of angle A.
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- 17 -
Applications of Linear Equations Notes
Example 3-5. Finding the Measure of an Angle
Twice the complement of angle A is 40° less than the supplement of angle A. Find the
measure of angle A.
Example 3-5. Finding the Measure of an Angle
Homework on Type 3
Finish examples not completed in class and Barron’s Book p. 48 (28, 29, 36). You must
show all of your work. Answers just written in the Barron’s Book are not acceptable.
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- 18 -
Applications of Linear Equations Notes
- 19 -
Read and answer the following questions.
Homework on Type 4:
Barron’s Book p.46 (3, 30, 31, 33, 34, 35) You must show all of your work. Answers
just written in the Barron’s Book are not acceptable.
Type 4. Finding Consecutive Integers
Two integers the differ by 1 are called consecutive integers. For example, 3 and 4 are
consecutive integers.
Write your own example of a pair of consecutive integers.
An example of consecutive negative integers is -2 and -1. Write your own example of a
pair of negative consecutive integers.
In general, if x represents an integer,
+ 1 represents the next larger consecutive integer.
Consecutive even integers, such as 8 and 10, differ by 2. Similarly consecutive odd
integers, such as 9 and 11, also differ by two. In general, if x represents an even integer,
+ 2 represents the next larger even consecutive integer. In general, if x represents an
odd integer, + 2 represents the next larger consecutive odd integer.
Problem Solving Hint
When solving consecutive integer problems, if
Two consecutive integers, use , + 1
Two consecutive even integers, use , + 2
Two consecutive odd integers, use , + 2
=the first integer, then for any
Three consecutive integers, use , + 1, + 2
Three consecutive even integers, use , + 2, + 4
Three consecutive odd integers, use , + 2, + 4
Example 4-1.
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Applications of Linear Equations Notes
- 20 -
Two pages that face each other in this book have 277 as the sum of their page numbers.
What are the page numbers?
Step 1 Read the problem carefully. Because the two pages face each other, they must
have page numbers that are consecutive integers.
Step 2 Assign a variable.
Let…
x=
the first page number
+1=
the next page number
Step 3 Write an equation.
First page
x
and
+
Second
page
x+1
is
=
Sum of
pages
277
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Applications of Linear Equations Notes
Step 4 Solve the equation.
- 21 -
x  x  1  277
2 x  1  277
2 x  1  277
1
1
2 x  278
2 x 278

2
2
= 138
Step 5 State the answer.
x=
the first page number
+1 =
Combine terms
Subtract 1
Combine terms
Divide by 2
x = 138
+ 1 = 139
the next page number
The first page number is 138, and the next page number is 138+1=139.
Step 6 Check. The sum of 138 and 139 is 277. The answer is correct
Example 4-2.
If the smaller of two consecutive odd integers is doubled, the result is 7 more than the
larger of the two integers. Find the two integers.
Step 1 Read the problem carefully. Find the two integers.
Step 2 Assign a variable.
Let…
x=
the smaller integer
+2=
the next odd integer
Step 3 Write an equation.
If the
smaller is
doubled
2(x)
the
result
is
=
7
7
More
than
+
The
larger
(x+2)
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Applications of Linear Equations Notes
Step 4 Solve the equation.
- 22 -
2x   7  x  2
2x  9  x
2x  9  x
x
x
Subtract x
Combine terms
x9
Step 5 State the answer.
x=
the smaller integer
+2=
the next odd integer
The first integer is 9 and the second is 9+2=11..
Combine terms
x =9
+ 2 =9+2=11
Step 6 Check. The when 9 is doubled, we get 18, which is 7 more than the larger odd
integer, 11. The answer is correct.
Answer the following questions on your own paper. Show all work.
1. The numbers on two consecutively numbered gym lockers have a sum of 137. What
are the locker numbers?
2. The sum of two consecutive checkbook check numbers is 357. Find the numbers.
3. Two pages that are back-to-back in this book have 293 as the sum of their page
numbers. What are the page numbers?
4. Find two consecutive odd integers such that twice the larger is 17 more than the
smaller.
5. Find two consecutive even integers such that the smaller added to three times the
larger gives a sum of 46.
6. Two houses on the same side of the street have house numbers that are consecutive
even integers. The sum of the integers is 58. What are the two house numbers?
7. When the smaller of two consecutive integers is added to three times the larger, the
result is 43. Find the integers.
________________________________________________________________________
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Applications of Linear Equations Notes
8. If five times the smaller of two consecutive integers is added to three times the larger,
the result is 59. Find the integers.
Solutions:
1. 68, 69
2. 178, 179
3. 146, 147
4. 13, 15
5. 10, 12
6. 28, 30
7. 10, 11
8. 7,8
Homework on Type 4:
Barron’s Book p.46 (3, 30, 31, 33, 34, 35) You must show all of your work. Answers
just written in the Barron’s Book are not acceptable.
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- 23 -