How to Solve: Linear Equations Word Problems

How to Solve: Linear Equations Word Problems

Introduction

"Master linear equations word problems, and you’ll crack everything from budgeting your first paycheck to decoding SAT, GCSE, or AP exam questions in under 3 minutes—guaranteed."

What You Need To Know First

Before diving in, ensure you understand: 1. Solving one-step and two-step linear equations (e.g., 3x + 5 = 20). 2. Translating words into algebraic expressions (e.g., "5 more than a number" → x + 5). 3. Basic arithmetic operations (addition, subtraction, multiplication, division).

If any of these feel shaky, pause and review them first—this guide builds on them.

Key Vocabulary

Formulas To Know

Step-by-Step Method

Follow these 6 steps for every word problem. No exceptions.

1. Read the problem twice.
2. First read: Understand the story.

3. Second read: Circle numbers, underline key phrases (e.g., "total," "twice as much," "consecutive").

4. Define the variable.

5. Ask: "What am I solving for?"

6. Write: "Let x = [what the variable represents]." (e.g., Let x = the number of tickets sold.)

7. Translate words into an equation.

8. Convert phrases to math:
   • "5 more than a number" → x + 5
   • "3 less than twice a number" → 2x – 3
   • "The sum of two numbers is 12" → x + y = 12

9. Write the full equation.

10. Solve the equation.

11. Use inverse operations (undo addition with subtraction, etc.).

12. Show every step (examiners award method marks!).

13. Check your answer.

14. Plug the solution back into the original problem (not just the equation).

15. Ask: "Does this make sense in the real-world context?"

16. Write a full-sentence answer.

17. Example: "The number of adult tickets sold was 24."

Worked Example Using the Steps

Problem: A gym charges a $30 sign-up fee plus $15 per month. If Sarah paid a total of $120, how many months has she been a member?

Step 1: Read twice. - Key info: $30 fee, $15/month, total paid = $120.

Step 2: Define the variable. - Let m = the number of months Sarah has been a member.

Step 3: Translate to an equation. - Total cost = Sign-up fee + (Monthly fee × Number of months) - 120 = 30 + 15m

Step 4: Solve. - Subtract 30 from both sides: 120 – 30 = 15m → 90 = 15m - Divide by 15: m = 90 ÷ 15 → m = 6

Step 5: Check. - $30 + ($15 × 6) = $30 + $90 = $120 ✔️

Step 6: Full-sentence answer. - "Sarah has been a member for 6 months."

Worked Examples

Example 1 – Basic

Problem: The sum of three consecutive integers is 72. Find the integers.

Step 1: Read twice. - Key info: 3 consecutive integers, sum = 72.

Step 2: Define the variable. - Let n = the first integer. - Then the next two integers are n + 1 and n + 2.

Step 3: Translate to an equation. - Sum: n + (n + 1) + (n + 2) = 72

Step 4: Solve. - Combine like terms: 3n + 3 = 72 - Subtract 3: 3n = 69 - Divide by 3: n = 23 - So the integers are 23, 24, 25.

Step 5: Check. - 23 + 24 + 25 = 72 ✔️

Step 6: Full-sentence answer. - "The three consecutive integers are 23, 24, and 25."

What we did and why: - Used n, n+1, n+2 for consecutive integers. - Combined like terms to simplify the equation.

Example 2 – Medium

Problem: A taxi charges a $5 base fare plus $2 per kilometer. If a ride costs $23, how far was the trip?

Step 1: Read twice. - Key info: $5 base fare, $2/km, total cost = $23.

Step 2: Define the variable. - Let k = the number of kilometers traveled.

Step 3: Translate to an equation. - Total cost = Base fare + (Cost per km × Kilometers) - 23 = 5 + 2k

Step 4: Solve. - Subtract 5: 18 = 2k - Divide by 2: k = 9

Step 5: Check. - $5 + ($2 × 9) = $5 + $18 = $23 ✔️

Step 6: Full-sentence answer. - "The trip was 9 kilometers long."

What we did and why: - Recognized the "base fee + variable cost" structure. - Isolated the variable (k) to find the distance.

Example 3 – Exam Style

Problem: Liam is 5 years older than twice his sister’s age. If the sum of their ages is 32, how old is Liam?

Step 1: Read twice. - Key info: Liam’s age = 5 + 2 × (sister’s age); sum of ages = 32.

Step 2: Define the variable. - Let s = sister’s age. - Then Liam’s age = 2s + 5.

Step 3: Translate to an equation. - Sum of ages: s + (2s + 5) = 32

Step 4: Solve. - Combine like terms: 3s + 5 = 32 - Subtract 5: 3s = 27 - Divide by 3: s = 9 - Liam’s age = 2(9) + 5 = 23

Step 5: Check. - Sister = 9, Liam = 23; sum = 9 + 23 = 32 ✔️

Step 6: Full-sentence answer. - "Liam is 23 years old."

What we did and why: - Defined the sister’s age first (easier variable). - Substituted back to find Liam’s age after solving for s.

Common Mistakes

Exam Traps

1-Minute Recap

[Camera on. Speak directly to the student.]

"Alright, it’s the night before your exam, and you’ve got 60 seconds to lock this in. Here’s the deal:

1. Read twice. Circle numbers, underline key phrases.
2. Define your variable. Let x = what you’re solving for. Be specific.
3. Translate words to math. "Twice a number" = 2x. "5 less than that" = 2x – 5.
4. Solve step by step. Show your work—examiners give marks for method.
5. Check your answer. Plug it back into the problem (not just the equation).
6. Write a full-sentence answer. No naked numbers!

Pro tip: If the problem mentions "consecutive numbers," use n, n+1, n+2. If it’s about money, use Total = Fixed + Variable × Quantity.

You’ve got this. Now go ace that exam!

[Camera off.]