Two Step Equations Word Problems Worksheet

Solving Two-Step Equation Word Problems: A Comprehensive Guide with Worksheets

This article provides a comprehensive guide to solving two-step equation word problems, a crucial skill in algebra. We'll break down the process step-by-step, offering practical examples, and providing you with worksheets to practice. Mastering two-step equations is key to unlocking more complex mathematical concepts. This guide is designed for students of all levels, from those just beginning to grasp the concept to those looking to solidify their understanding.

Understanding Two-Step Equations

Before diving into word problems, let's refresh our understanding of two-step equations. A two-step equation is an algebraic equation that requires two steps to solve for the unknown variable (usually represented by x or another letter). These equations typically involve addition, subtraction, multiplication, and/or division. A simple example is: 2x + 5 = 11. To solve this, we first subtract 5 from both sides (2x = 6), and then divide both sides by 2 (x = 3).

The core principle behind solving any equation is maintaining balance. Whatever operation you perform on one side of the equation, you must perform the same operation on the other side to keep the equation true.

From Words to Equations: Deciphering Word Problems

The challenge with word problems lies in translating the written description into a mathematical equation. This requires careful reading and identifying the key information:

• The Unknown: What are we trying to find? This will be your variable (e.g., x, y, n).
• The Operations: What mathematical operations (addition, subtraction, multiplication, division) are described in the problem?
• The Relationships: How are the different quantities related to each other?

Step-by-Step Guide to Solving Two-Step Equation Word Problems

Let's break down the process into manageable steps:

1. Read and Understand: Carefully read the problem multiple times. Underline or highlight key information. Identify the unknown quantity you need to find.

2. Define the Variable: Assign a variable (usually x) to represent the unknown quantity.

3. Translate into an Equation: Translate the words into a mathematical equation. This is often the most challenging step. Look for keywords:

   • Addition: "more than," "increased by," "sum," "total"
   • Subtraction: "less than," "decreased by," "difference," "minus"
   • Multiplication: "times," "product," "of"
   • Division: "divided by," "quotient," "per"

4. Solve the Equation: Use inverse operations to isolate the variable and solve for its value. Remember to maintain balance throughout the process.

5. Check Your Answer: Substitute your solution back into the original equation to ensure it makes the equation true. Does the solution make sense in the context of the word problem?

Example Word Problems and Solutions

Let's work through some examples to illustrate the process:

Example 1:

Problem: Sarah bought 3 books and a magazine. Each book cost $8, and the magazine cost $5. If Sarah spent a total of $29, how much did each book cost?

1. Read and Understand: We need to find the cost of each book.

2. Define the Variable: Let x represent the cost of each book.

3. Translate into an Equation: The total cost is 3 books at x dollars each plus the magazine cost. So the equation is: 3x + 5 = 29

4. Solve the Equation:

   • Subtract 5 from both sides: 3x = 24
   • Divide both sides by 3: x = 8

5. Check Your Answer: 3(8) + 5 = 29. This is correct. Each book cost $8.

Example 2:

Problem: John had 25 apples. He gave some apples to his friends, and then he bought 10 more. Now he has 18 apples. How many apples did he give away?

1. Read and Understand: We need to find the number of apples John gave away.

2. Define the Variable: Let x represent the number of apples John gave away.

3. Translate into an Equation: He started with 25, gave away x, and then added 10. So the equation is: 25 - x + 10 = 18

4. Solve the Equation:

   • Combine like terms: 35 - x = 18
   • Add x to both sides and subtract 18 from both sides: x = 17

5. Check Your Answer: 25 - 17 + 10 = 18. This is correct. John gave away 17 apples.

Example 3 (More Challenging):

Problem: A rectangle has a length that is 3 cm more than twice its width. The perimeter of the rectangle is 42 cm. Find the length and width of the rectangle.

1. Read and Understand: We need to find the length and width.

2. Define the Variable: Let w represent the width. The length is then 2w + 3.

3. Translate into an Equation: The perimeter of a rectangle is 2(length + width). So the equation is: 2(2w + 3 + w) = 42

4. Solve the Equation:

   • Simplify the parentheses: 2(3w + 3) = 42
   • Distribute the 2: 6w + 6 = 42
   • Subtract 6 from both sides: 6w = 36
   • Divide both sides by 6: w = 6
   • Now find the length: 2(6) + 3 = 15

5. Check Your Answer: 2(15 + 6) = 42. This is correct. The width is 6 cm and the length is 15 cm.

Worksheet 1: Basic Two-Step Equation Word Problems

1. A number increased by 7 is 15. What is the number?

2. Twice a number decreased by 5 is 9. What is the number?

3. Three times a number plus 4 is 22. What is the number?

4. A number divided by 4 and then increased by 2 is 7. What is the number?

5. Ten less than four times a number is 6. What is the number?

Worksheet 2: More Challenging Two-Step Equation Word Problems

1. Maria bought 5 pencils and a notebook. The pencils cost $1 each, and the total cost was $9. How much did the notebook cost?

2. A rectangular garden has a length that is 5 feet longer than its width. The perimeter of the garden is 38 feet. Find the length and width of the garden.

3. John earns $12 per hour. He worked a certain number of hours and earned $96. Then he received a bonus of $20. How many hours did he work?

4. A school bus has a certain number of students. 12 students got off at the first stop, and then 15 students got on. Now there are 27 students on the bus. How many students were originally on the bus?

5. The sum of three consecutive even numbers is 48. What are the three numbers?

Frequently Asked Questions (FAQ)

Q: What if I get a negative answer?

A: Negative answers are perfectly acceptable in algebra. Just make sure your solution makes sense within the context of the word problem. For example, a negative number wouldn't make sense if it represents the number of apples.

Q: How can I improve my ability to translate word problems into equations?

A: Practice is key! Work through many different types of word problems. Try to break down each problem into smaller, more manageable parts. Focus on identifying the key information and relationships between the quantities.

Q: What if I make a mistake?

A: Don't worry! Mistakes are part of the learning process. Carefully review your work, check your calculations, and try again. Understanding where you went wrong is just as important as getting the right answer.

Conclusion

Solving two-step equation word problems is a fundamental skill in algebra. By following the step-by-step process outlined in this article and practicing regularly using the provided worksheets, you will build confidence and proficiency in tackling these problems. Remember to focus on understanding the problem, translating it into an equation, and carefully checking your solution. With dedicated practice, you'll master this important skill and be well-prepared for more advanced algebraic concepts. Good luck!