Download Applications Using Linear Equations This section will

Applications Using Linear Equations
This section will discuss several types of applications. No
matter how simple or complex the application is, the following
steps will help to translate and solve the problem.
1.
2.
3.
4.
5.
Read through the entire problem
Organize the information
Write the equation
Solve the equation
Check the answer
Direct Translation
With this type of application, you will translate word to symbol.
Look for key words and phrases. The following table gives
some common translations.
Word or Phrase
is
the same as
sum
more than
difference
less than
product
of
Symbol
=
=
+
+
×
×
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 1:
The sum of four and a number is ten.
The sum of four and a number is ten.
4 + x = 10
-4
-4
x= 6
Subtract 4 from both sides
Check: The sum of 4 and 6 is 10.
Or 4+6=10.
Example 2:
If 28 less than five times a number is 232. What is the number?
If 28 less than five times a number is 232.
5x – 28 = 232
+ 28 + 28 Add 28 to both sides
5x
= 260
5
5 Divide both sides by 5
x
= 52
Check: 28 less than 5(52) is 232.
Or 5(52) – 28 = 232
260 – 28 = 232
232 = 232
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Consecutive Integers
Consecutive integers are numbers that come one after the other,
such as 3, 4, 5. To get from one number to the next, we only
have to add 1, so if the first number is x, the next number is x+1.
Consecutive odd or even integers are numbers that are spaced
apart by two, such as 2, 4, 6 or 3, 5, 7. If the first number is x,
the next number is x+2.
Example 3:
The sum of two consecutive integers is 93. What are the integers?
The first number is x
The second number is x+1
x + x + 1 = 93
2x + 1 = 93 Combine like terms x + x
- 1 - 1 Subtract 1 from both sides
2x
= 92
2
2
Divide both sides by 2
x
= 46
The first number x = 46
The second number x+1 = 46+1 = 47
Check: 47 comes after 46.
Also 46+47=93.
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 4:
The sum of three consecutive even integers is 246. What are the
integers?
The first number is x
The second number is x+2
The third number is x+4
x + x + 2 + x + 4 = 246
3x + 6 = 246 Combine x + x + x and 2 + 4
-6
- 6 Subtract 6 from both sides
3x
= 240
3
3 Divide both sides by 3
x
= 80
The first number x = 80
The second number x+2 = 80+2 = 82
The third number x+4 = 80+4 = 84
Check: 80, 82, and 84 are consecutive even numbers.
Also 80 + 82 + 84 = 246.
Perimeter of a Shape
For applications involving perimeter (the distance around an
object), drawing a picture is helpful.
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 5:
The perimeter of a rectangle is 44 feet. The length is 5 feet less than
twice the width. Find the dimensions.
2w – 5
w
There are two widths of the same size, w+w or 2w
There are two lengths of the same size, (2w-5)+(2w-5) or 2(2w-5)
2w + 2(2w – 5) = 44
2w + 4w – 10 = 44
6w – 10 = 44
+ 10 + 10
6w
= 54
6
6
w
= 9
Distribute the 2
Combine the like terms 2w + 4w
Add 10 to both sides
Divide both sides by 6
The width w = 9 ft
The length 2w-5 = 2(9)-5 = 13 ft
Check: Length + Length + Width + Width = 9 + 9 + 13 + 13 = 44
Or 2L + 2W = 2(9) + 2(13) = 18 + 26 = 44
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 6:
If one side of a triangle is twice as long as the shortest side, and the third
side is 3 times as long as the shortest side, find the dimensions of a
triangle whose perimeter is 72 meters.
2x
x
3x
2x + 3x + x = 72
6x = 72
6
6
x = 12
Combine like terms 2x + 3x + x
Divide both sides by 6
The shortest side x = 12 m
The side twice as long as the shortest 2x = 2(12) = 24 m
The side 3 times as long as the shortest 3x = 3(12) = 36 m
Check: 2(12) + 3(12) + 12 = 24 + 36 + 12 = 72
Basic “Real Life” Applications
The following are examples of a type of application that is very
common. They are similar to the above examples but may seem
slightly different. When organizing these problems, it is
important to clearly identify the variable.
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 7:
A sofa and loveseat together costs $444. The cost of the sofa is double
the cost of the loveseat. How much do they cost?
The love seat is x
The sofa is double the love seat, so 2x
x + 2x = 444
3x = 444
3
3
x = 148
Combine like terms x + 2x
Divide both sides by 3
The love seat x = $148
The sofa 2x = 2(148) = $296
Check: $148 + $296 = $444
Example 8:
An eight foot board is cut into two pieces. One piece is 2 feet longer
than the other piece. How long are the pieces?
x
= 8 feet
x+x+2=8
2x + 2 = 8
-2 -2
2x
=6
2
2
x
=3
Combine like terms x + x
Subtract 2 from both sides
Divide both sides by 2
The short piece x = 3 ft
The long piece x + 2 = 3 + 2 = 5 ft
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Basic Percent Problems
There are three basic types of percent problems:
1. Find a given percent of a given number.
For example: find 25% of 640.
2. Find a percent given two numbers.
For example: 15 is what percent of 50?
3. Find a number that is a given percent of another number.
For example: 10% of what number is 12?
Example 9:
What number is 25% of 640?
Translate the words into an equation.
What number is 25% of 640?
x
= 25% × 640
x = .25 × 640
x = 160
Change 25% to the decimal .25
160 is 25% of 640
Example 10:
15 is what percent of 50?
Translate the words into an equation.
15 is what percent of 50?
15 =
x
× 50
15 = 50x
50 50
Divide both sides by 50
x = 0.30 = 30% Change 0.30 to the percent 30%
15 is 30% of 50
Modified from Prealgebra Textbook, by the Department of Mathematics, College of the Redwoods, CCBY 2009. Licensed under a Creative Commons Attribution 3.0 Unported License
(http://creativecommons.org/licenses/by-nc-sa/3.0)
Example 11:
10% of what number is 12?
Translate the words into an equation.
10% of what number is 12?
10% ×
x
= 12
.10x = 12
Change 10% to the decimal .10
(100).10x = (100)12 Multiply by 100 to clear the decimal
10x = 1200
10
10
Divide both sides by 10
x = 120
10% of 120 is 12
Modified from Prealgebra Textbook, by the Department of Mathematics, College of the Redwoods, CCBY 2009. Licensed under a Creative Commons Attribution 3.0 Unported License
(http://creativecommons.org/licenses/by-nc-sa/3.0)