Grade 8 Linear Equations: Word Problems

Real-world linear equations

What You'll Learn

Set up a linear equation from a story
Solve and interpret the solution in context

📖Translate, then solve

Let a variable stand for the unknown the question asks about. Turn each sentence into an expression: a one-time fee is a constant, a cost per month or per visit multiplies the variable, and "the same as" or "in all" becomes the equals sign. Then solve with inverse operations.

✏️Example

A gym charges a $25 sign-up fee plus $15 per month. Maria has paid $160. Let m be the months: 25 + 15m = 160, so 15m = 135 and m = 9 months.

💡Two plans that meet

When two plans, savers or plants change at different rates, set their expressions equal and collect the variable on one side: 40 + 5v = 10v gives 40 = 5v, so v = 8.

Worked examples

Real problems from the practice worksheets, solved one step at a time.

Example 1
At 7 a.m. the temperature was −10°F. It rose 4°F every hour. After how many hours did it reach 18°F?
1. Let h be the number of hours. Starting at −10°F and rising 4°F each hour, the temperature is −10 + 4h. It reaches 18°F when −10 + 4h = 18.
2. Add 10 to both sides: 4h = 18 + 10 = 28.
3. Divide both sides by 4: h = 28 ÷ 4 = 7.
4. Check: put h = 7 back in. −10 + 4 × 7 = 18 ✓
5. So h = 7, and the answer is 7 hours.
Answer: 7
Example 2
A gym charges a one-time $50 sign-up fee plus $22 per month. Maria has paid $314 in all. How many months has she been a member?
1. Let m be the number of months. The $50 fee plus $22 for each of m months costs 50 + 22m dollars, and Maria paid $314 in all: 50 + 22m = 314.
2. Subtract 50 from both sides: 22m = 314 − 50 = 264.
3. Divide both sides by 22: m = 264 ÷ 22 = 12.
4. Check: put m = 12 back in. 50 + 22 × 12 = 314 ✓
5. So m = 12, and the answer is 12 months.
Answer: 12
Example 3
A taxi charges $3.00 to start a ride plus $2.25 for each mile. A ride cost $27.75. How many miles long was the ride?
1. Let x be the number of miles. $3.00 to start plus $2.25 for each of x miles costs 3 + 2.25x dollars, and the ride cost $27.75: 3 + 2.25x = 27.75.
2. Subtract 3 from both sides: 2.25x = 27.75 − 3 = 24.75.
3. Divide both sides by 2.25: x = 24.75 ÷ 2.25 = 11.
4. Check: put x = 11 back in. 3 + 2.25 × 11 = 27.75 ✓
5. So x = 11, and the answer is 11 miles.
Answer: 11