Lesson 4.2: Forming and Solving Equations from Problems

Introduction

In this lesson, we will explore how to translate worded problems into linear equations and solve them. Our objectives for this lesson are to:

• Understand how to translate a worded situation into a linear equation.
• Solve the equation and interpret the answer in the context of the problem.
• Check that the answer is sensible for the original problem.
• Formulate a linear equation from a worded problem.
• Solve the equation and state the answer in context with units.

Understanding how to turn a real-world problem into a mathematical equation is a vital skill. It allows us to utilize algebra to find solutions to everyday situations. Let’s dive into this topic by breaking down the process step by step.

Translating Worded Situations into Linear Equations

Translating a worded problem involves identifying the unknowns, determining the relationships between these unknowns, and representing them using algebraic expressions and equations. The key steps are:

1. Identify the Unknowns: Assign variables to the unknown quantities.
2. Read the Problem Carefully: Look for key terms and phrases that indicate mathematical operations.
3. Set Up the Equation: Use the relationships and operations identified to write a linear equation.

Example 1: A Simple Problem

Problem: John has $20. He buys some candy, which costs $3 each. After buying the candy, he has $8 left. How many candies did he buy?

1. Identify the Unknowns: Let $ x $ be the number of candies John buys.
2. Read the Problem: We know the total amount John starts with, how much he spends on candy, and how much he has left.
3. Set Up the Equation: $ 20 - 3x = 8 $

This represents the initial amount ($20), minus the total cost of the candy ($3 for each candy he buys), which equals the amount he has left ($8).

Solving the Equation

Now, let’s solve the equation step by step.

1. Start with the equation:

$ 20 - 3x = 8 $

1. Subtract 20 from both sides:

$ -3x = 8 - 20 $

$ -3x = -12 $

1. Divide both sides by -3:

$ x = 4 $

Interpretation: John bought 4 candies.

Example 2: A More Detailed Problem

Problem: A group of friends went out to a restaurant. They spent a total of $120 on food and drinks. If each person paid $15, how many friends were there?

1. Identify the Unknowns: Let $ y $ be the number of friends.
2. Read the Problem: The total cost is $120, and each person paid $15.
3. Set Up the Equation: $ 15y = 120 $

Solving the Equation

Now let’s solve the equation.

1. Start with the equation:

$ 15y = 120 $

1. Divide both sides by 15:

$ y = \frac{120}{15} $

$ y = 8 $

Interpretation: There were 8 friends in the group.

Checking Our Answers

It is crucial to verify that our answers make sense in the context of the original problems. This helps ensure we have formulated the equations correctly and interpreted them properly.

Example Check

In the first example, we found that John bought 4 candies. We can check that:

• Cost of candies: $ 3 \times 4 = 12 $
• Total spent: $ 20 - 12 = 8 $ (which matches the problem statement).

This confirms our solution is correct.

Handling More Complex Problems

Some problems may involve multiple steps or more than one unknown. Let’s look at a problem requiring two equations.

Example 3: Simultaneous Equations

Problem: Sarah and Tom are together preparing for a party. Sarah has 3 times as many balloons as Tom. If together they have 48 balloons, how many balloons does each of them have?

1. Identify the Unknowns: Let $ a $ be the number of balloons Tom has, and let $ s $ be the number of balloons Sarah has.
2. Set Up the Equations: From the problem, we can write two equations:

$ s = 3a $

$ s + a = 48 $

Solving the Simultaneous Equations

We can substitute the first equation into the second:

1. Substitute $ s $ in the second equation:

$ 3a + a = 48 $

$ 4a = 48 $

1. Divide by 4:

$ a = 12 $

1. Use $ a $ to find $ s $:

$ s = 3a = 3 \times 12 = 36 $

Interpretation: Tom has 12 balloons, and Sarah has 36 balloons.

Conclusion

In this lesson, we learned the importance of translating worded problems into equations. We practiced forming linear equations, solving them, and checking our answers in context. This skill is essential for solving real-world problems using algebra.

Study Notes

• To form a linear equation from a problem, identify unknowns and relationships.
• Solve the equation systematically.
• Always interpret the answer in the context of the original problem.
• Check your answer for accuracy and sensibility.
• For problems with multiple unknowns, use simultaneous equations as needed.