Lesson 102: Word Problems Using Variables & Equations

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• Lesson Description
• Learning Objectives
• Topics Covered
• Key Terms
• Lesson Examples
• Practice Problems
• Tips & Common Mistakes
• Summary
• Challenge Problems
• Unit Navigation

Lesson Description

This lesson teaches students how to solve word problems by identifying unknowns, assigning variables, and forming equations. Using algebraic methods, students learn to model real-world situations and systematically find solutions.

Learning Objectives

• Identify unknown quantities in word problems and assign appropriate variables.
• Translate word problems into algebraic equations.
• Solve equations to find the value of the unknown and interpret the result in context.

Topics Covered

• Defining variables from word problems
• Setting up linear equations
• Solving equations and checking solutions
• Interpreting solutions in real-world context

Key Terms

• Variable: A symbol (like x or y) that represents an unknown number.
• Equation: A mathematical statement showing two expressions are equal.
• Solution: The value of the variable that makes the equation true.

Lesson Examples

Example 1: Buying Tickets

A movie theater sells adult tickets for $12 and child tickets for $8. If 5 tickets were sold for a total of $48, how many adult tickets were sold?

1. Let x = number of adult tickets sold.
2. Then 5 − x = number of child tickets sold.
3. Set up the equation: 12x + 8(5 − x) = 48.
4. Solve: 12x + 40 − 8x = 48 → 4x = 8 → x = 2.
5. Answer: 2 adult tickets were sold.

Example 2: Age Problem

Sarah is 4 years older than twice Tom's age. If their combined ages are 28, how old is Tom?

1. Let x = Tom's age.
2. Sarah's age = 2x + 4.
3. Equation: x + (2x + 4) = 28 → 3x + 4 = 28.
4. Solve: 3x = 24 → x = 8.
5. Answer: Tom is 8 years old; Sarah is 20 years old.

Practice Problems

1. A bookstore sold 30 books in one day. If fiction books cost $10 and non-fiction books cost $15, and total sales were $360, how many fiction books were sold?
2. A number is 7 more than twice another number. If their sum is 29, find both numbers.
3. Jenny bought pens for $1 each and notebooks for $3 each. She bought 10 items and spent $18. How many pens did she buy?
4. A train travels x miles per hour. If it travels for 3 hours and covers 180 miles, find x.

Tips & Common Mistakes

• Carefully define what each variable represents before setting up the equation.
• Write all parts of the problem as expressions before combining them into an equation.
• Always check that the solution makes sense in the context of the problem.

Summary

Solving word problems with variables requires identifying unknowns, translating the situation into an equation, and solving carefully. Verifying that the solution fits the context ensures accuracy and builds confidence in problem-solving.

Challenge Problems

1. A store sells pencils for $0.50 each and erasers for $0.75 each. If a customer buys 20 items for $13, how many of each item did they buy?
2. Two numbers differ by 5. Three times the smaller number plus twice the larger number is 27. Find both numbers.
3. A rectangle's length is 3 meters longer than its width. If the perimeter is 26 meters, find the dimensions.