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
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