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 set up and solve consecutive integer word problems

## How to Solve a Consecutive Integer Word Problem

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.

Some Consecutive Integers:

1,2

3,4

5,6

7,8

9,10

Let's take a look at an example.

Example 1: Solve each word problem

When we add three consecutive integers together, the result is 21. What are the integers?

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

- Read the problem carefully and determine what you are asked to find
- Write down the main objective of the problem

- Assign a variable to represent the unknown
- If more than one unknown exists, we express the other unknowns in terms of this variable

- Write out an equation which describes the given situation
- Solve the equation
- State the answer using a nice clear sentence
- 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.

### 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 1: Solve each word problem

When we add three consecutive integers together, the result is 21. What are the integers?

- Read the problem carefully and determine what you are asked to find
- We are asked to find three consecutive integers.

- 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

- Write out an equation which 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

- 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

- 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.

- 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.

#### Skills Check:

Example #1

Solve each word problem.

The sum of two pages that face each other in a book is 177. What are the page numbers?

Please choose the best answer.

A

86, 91

B

85, 92

C

176, 177

D

89, 90

E

88, 89

Example #2

Solve each word problem.

The sum of three consecutive even integers is 102. Find the integers.

Please choose the best answer.

A

32, 36, 38

B

20, 40, 62

C

18, 24, 40

D

32, 34, 36

E

2, 40, 60

Example #3

Solve each word problem.

When the smaller of two consecutive integers is added to 3 times the larger, the result is -37. Find the two integers.

Please choose the best answer.

A

7, 5

B

-1, 3

C

-6, -5

D

-10, -9

E

-12, -11

Congrats, Your Score is 100%

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