Word Problems Involving Linear Equations

Unit: Linear Equations in One Variable

Word Problems Involving Linear Equations

Word problems involving linear equations require translating a real-world situation into a mathematical equation and then solving for the unknown variable. These problems often involve relationships between quantities, rates, and times.

Steps to Solve Word Problems

1. Read the Problem Carefully:
   • Identify what you are asked to find.
   • Determine the information given.
2. Define the Variables:
   • Choose a variable to represent the unknown quantity.
   • Write down what the variable represents.
3. Set Up the Equation:
   • Translate the words into a mathematical equation using the defined variable.
4. Solve the Equation:
   • Use algebraic methods to solve for the variable.
5. Check the Solution:
   • Substitute the solution back into the original context to ensure it makes sense.
   • Verify the solution meets all conditions of the problem.
6. Answer the Question:
   • Write a complete sentence answering the problem’s question.

Examples

1. Basic Example:
   • Problem: A number increased by 7 is 15. What is the number?
   • Solution:
     • Let x be the number.
     • Equation: x+7=15
     • Solve: x=15−7=8
     • Answer: The number is 8.
2. Age Problems:
   • Problem: John is 3 years older than his sister. If the sum of their ages is 23, how old is each?
   • Solution:
     • Let x be the sister’s age.
     • John’s age: x+3
     • Equation: x+(x+3)=23
     • Solve: 2𝑥+3=23
     • 2x=20
     • x=10
     • Sister’s age: 10
     • John’s age: 10+3=13
     • Answer: The sister is 10 years old and John is 13 years old.
3. Distance-Rate-Time Problems:
   • Problem: A car travels at 60 miles per hour. How long will it take to travel 180 miles?
   • Solution:
     • Let t be the time in hours.
     • Equation: 60t=180
     • Solve: 𝑡==3
     • Answer: It will take 3 hours to travel 180 miles.
4. Mixture Problems:
   • Problem: How many liters of a 10% salt solution must be mixed with 20 liters of a 25% salt solution to get a 20% salt solution?
   • Solution:
     • Let x be the liters of the 10% solution.
     • Salt from 10% solution: 0.10x
     • Salt from 25% solution: 0.25×20=5
     • Total salt: 0.20(x+20)
     • Equation: 0.10x+5=0.20(x+20)
     • Solve: 0.10𝑥+5=0.20𝑥+4
     • 5−4=0.20𝑥−0.10𝑥
     • 1=0.10x
     • x=10
     • Answer: 10 liters of the 10% salt solution are needed.
5. Investment Problems:
   • Problem: Maria invests $5000 in two accounts. One account pays 5% interest and the other pays 7%. If the total interest earned is $320, how much is invested in each account?
   • Solution:
     • Let x be the amount invested at 5%.
     • Amount at 7%: 5000−x
     • Equation: 0.05𝑥+0.07(5000−𝑥)=320
     • Solve: 0.05𝑥+350−0.07𝑥=320
     • −0.02𝑥+350=320
     • −0.02𝑥=−30
     • 𝑥=1500
     • Amount at 5%: $1500
     • Amount at 7%: 5000−1500=3500
     • Answer: $1500 is invested at 5% and $3500 at 7%.

Summary

• Steps to Solve: Read carefully, define variables, set up the equation, solve, check, and answer.
• Examples: Include basic number problems, age problems, distance-rate-time problems, mixture problems, and investment problems.
• Key Skills: Translating words into equations, solving equations, and interpreting solutions in context.

Mastering word problems involves practice and the ability to connect mathematical concepts to real-life scenarios.