Lesson Objectives
• Demonstrate an understanding of how to solve a Linear Equation in One Variable
• Learn the six-step process for solving any word problem that involves a Linear Equation in One Variable
• Learn how to check the solution for a word problem
• Learn how to use phrases to set up equations
• Learn how to set up and solve word problems that involve sums of quantities
• Learn how to set up and solve consecutive integer word problems
• Learn how to set up and solve age word problems

## How to Solve Word Problems with Linear Equations

Over the course of the last few lessons, we have learned how to solve a linear equation in one variable. In real life, our problems will not be given to us in such a manner. We will need to create an equation based on the context of the given situation. When we first learn how to set up and solve word problems, it can be a real challenge for many students. It is often helpful to follow a step-by-step method for solving word problems.

### Six-step method for Applications of Linear Equations in One Variable

1. Read the problem carefully and determine what you are asked to find
• Write down the main objective of the problem
2. Assign a variable to represent the unknown
• If more than one unknown exists, we express the other unknowns in terms of this variable
3. Write out an equation that describes the given situation
4. Solve the equation
5. State the answer using a nice clear sentence
6. Check the result by reading back through the problem
• We need to make sure the answer is reasonable. In other words, if asked how many miles were driven to the store, the answer shouldn't be (-3) as we can't drive a negative amount of miles.

### Solve a Word Problem Involving Unknown Numbers

Our section on word problems generally starts with translating basic algebraic phrases. There will be keywords and phrases which will translate into mathematical expressions with addition, subtraction, multiplication, and division. Let's take a look at the table below which shows common words and phrases used in algebra word problems:
sum ofless than
more thanminus
plusdecreased by
increased bysubtracted from
togethertake away
Multiplication & Division Phrases:
Multiplication: Division:
productquotient
timesdivided by
ofhalf of
multiplied byratio of
twiceshared evenly
tripledaverage
Let’s work through an example.
Example 1: Solve each word problem.
The product of 6, and a number decreased by 3, is 24. What is the number?
1. Read the problem carefully and determine what you are asked to find
• We are asked to find an unknown number.
2. Assign a variable to represent the unknown
• Let x = the unknown number
3. Write out an equation that describes the given situation
• product of - means to multiply
• decreased by - means subtraction
• The keyword "is" means equals
• If we put this together:
• 6(x - 3) = 24
• 6 is multiplied by a number decreased by 3. This is the left part of the equation. Notice the parentheses. These are very important, as we want 6 to be multiplied by the result of x - 3. Then we have the keyword "is", which means equals. Lastly, we have 24, which is the right side of the equation.
4. Solve the equation
• 6(x - 3) = 24
• 6x - 18 = 24
• 6x - 18 + 18 = 24 + 18
• 6x = 42
• 6/6 x = 42/6
• x = 7
5. State the answer using a nice clear sentence
• Our unknown number is 7.
6. Check the result by reading back through the problem
• Think about this in terms of the problem. The product of 6 and 7 decreased by 3, is 24
• 7 decreased by 3 is 4
• 6 • 4 = 24
• 24 = 24
• This tells us our solution is correct and 7 is our unknown number.

### Solving a Word Problem Involving Sums of Quantities

In many word problems, we know the sum of two or more quantities, but the individual amounts are unknown. Let's take a look at an example.
Example 2: Solve each word problem.
Two high school football teams, the Cougars and the Aztecs combined to win a total of 13 games during the 10-game regular season. If the Cougars won 5 more games than the Aztecs, how many games did each team win?
1. Read the problem carefully and determine what you are asked to find
• We are asked to find how many games each team (Cougars, Aztecs) won.
2. Assign a variable to represent the unknown
• Let x = the number of games won by the Cougars
• Then x - 5 = the number of games won by the Aztecs
3. Write out an equation that describes the given situation
• If we combine the number of games won by the Cougars (x) with the number of games won by the Aztecs (x - 5), the result is (=) 13.
• x + (x - 5) = 13
4. Solve the equation
• x + (x - 5) = 13
• 2x - 5 = 13
• 2x - 5 + 5 = 13 + 5
• 2x = 18
• 2/2 x = 18/2
• x = 9
5. State the answer using a nice clear sentence
• Since x represents the number of games won by the Cougars, we know that the Cougars won 9 games.
• Since the Aztecs won 5 less games, we subtract 9 - 5, to get 4. This is the number of games won by the Aztecs.
• During the regular season, the Cougars won 9 games and the Aztecs won 4 games.
6. Check the result by reading back through the problem
• We know that the combined wins for the two teams were 13.
• Check: 9 (games won by Cougars) + 4 (games won by Aztecs) = 13
• We also know that the Cougars won 5 more games than the Aztecs.
• Check: 9 (games won by Cougars) - 5 = 4 (games won by the Aztecs)
• Since both checks are true, we have the correct answer.

### Consecutive Integer Word Problems

A common word problem that involves "sums of quantities" is to find unknown consecutive integers. Two consecutive integers will differ by 1. As an example, 1 and 2 are consecutive integers, as are 3 and 4.
Some Consecutive Integers:
1,2
3,4
5,6
7,8
9,10
Let's take a look at an example.
Example 3: Solve each word problem.
When we add three consecutive integers together, the result is 21. What are the integers?
1. Read the problem carefully and determine what you are asked to find
• We are asked to find three consecutive integers.
2. Assign a variable to represent the unknown
• Let x = smallest consecutive integer
• Then x + 1 = the middle consecutive integer
• Then (x + 1) + 1 = the largest consecutive integer
3. Write out an equation that describes the given situation
• If we combine the three consecutive integers: x, (x + 1), (x + 1) + 1, our result is (equals) 21.
• x + (x + 1) + (x + 1) + 1 = 21
4. Solve the equation
• x + (x + 1) + (x + 1) + 1 = 21
• x + x + 1 + x + 1 + 1 = 21
• 3x + 3 = 21
• 3x + 3 - 3 = 21 - 3
• 3x = 18
• 3/3 x = 18/3
• x = 6
5. State the answer using a nice clear sentence
• Since x represents the smallest consecutive integer, we know the smallest consecutive integer is 6.
• The middle consecutive integer is 7, from (6 + 1).
• The largest consecutive integer is 8, from (7 + 1).
• Our three consecutive integers are 6, 7, and 8.
6. Check the result by reading back through the problem
• We know that the sum of the consecutive integers is 21.
• Check: 6 + 7 + 8 = 21
• 13 + 8 = 21
• 21 = 21
• Since our three consecutive integers sum to 21, our answer is correct.

### Age Word Problems

Age word problems are very common in our study of algebra. This type of word problem generally gives the sum of the ages of family members or friends and asks us to determine the individual ages. Let's take a look at an example.
Example 4: Solve each word problem.
Two siblings, Jamie and Steven have a combined age of 15. Steven is twice the age of Jamie. Find the age of each sibling.
1. Read the problem carefully and determine what you are asked to find
• We are asked to find the age of each sibling.
2. Assign a variable to represent the unknown
• Let x = Jamie's age
• Then 2x = Steven's age (since it's twice the age of Jamie)
3. Write out an equation that describes the given situation
• If we add Jamie's age (x) with Steven's age (2x), the result is 15.
• x + 2x = 15
4. Solve the equation
• x + 2x = 15
• 3x = 15
• 3/3 x = 15/3
• x = 5
5. State the answer using a nice clear sentence
• Since x represents Jamie's age, we know that Jamie is 5 years old.
• Since 2x represents Steven's age, we know that Steven is 2 • 5 or 10 years old
• Jamie is 5 years old and Steven is 10 years old.
6. Check the result by reading back through the problem
• We know that the sum of their ages is 15.
• Check: 5 + 10 = 15
• 15 = 15
• We know that Steven is twice the age of Jamie
• Check: 2 • 5 = 10
• 10 = 10
• Since both of our checks were true, we can say our answer is correct.

#### Skills Check:

Example #1

Solve each word problem.

The result of a number being increased by 6 and then multiplied by 8 is 104. Find the number.

A
14
B
7
C
9
D
20
E
1

Example #2

Solve each word problem.

Last week, a grocery store sold 8/5 as many cans of tomatoes as bags of chips. The total number of cans of tomatoes and bags of chips sold last week was 273. How many bags of chips were sold last week?

A
168
B
105
C
205
D
135
E
155

Example #3

Solve each word problem.

When five times the smaller of two consecutive integers is added to three times the larger, the result is 59. Find the smaller integer.

A
9
B
1
C
12
D
8
E
7