Unit 2 · Lesson 4 of 6

Word problems with linear equations

About 25 minutes · 5 steps · one idea: name the unknown, then translate

You'll be able to: turn a short story about numbers, prices, rectangles, or travel into an equation, solve it, and answer the question that was asked.

  1. 1 Warm-up
  2. 2 The idea
  3. 3 Examples
  4. 4 Practice
  5. 5 Wrap-up

Step 1 of 5Warm-up

Two quick questions from earlier lessons. Answer from memory first, then check.

Solve \(y = 3x - 6\) for \(x\).

Add 6 to both sides: \(y + 6 = 3x\).

Divide the whole side by 3: \(x = \dfrac{y + 6}{3}\).

Compute \(2 + 3(5 - 1)^2 \div 6\).

Parentheses first: \(5 - 1 = 4\). Then the exponent: \(4^2 = 16\).

Multiply and divide left to right: \(3 \cdot 16 = 48\), then \(48 \div 6 = 8\).

Add last: \(2 + 8 = 10\).

Step 2 of 5The idea

A word problem is an equation written in English. Your job is to translate it, one phrase at a time. The math part is the same as the last three lessons.

Five steps:

  1. Name the unknown: "Let \(x\) = the smaller number."
  2. Write every other amount using \(x\) (for example, "6 more" is \(x + 6\)).
  3. Translate the sentence: "is" means \(=\), "of" with a percent means times.
  4. Solve the equation.
  5. Answer the actual question, with units, and check it in the story.

Patterns to know: consecutive integers are \(x, x+1, x+2\); consecutive odd (or even) integers are \(x, x+2, x+4\). A price after \(r\%\) off is the original minus \(r\%\) of it. Rectangle perimeter: \(P = 2l + 2w\). Travel: distance = rate · time, \(d = rt\).

Why this shows up on the placement test: the test mixes in a few story problems, and the hard part is always the setup, not the solving.

Step 3 of 5Worked examples

Example 1: three consecutive integers add up to 72. What are they?

  1. Name: let \(x\) = the first integer. Then the next two are \(x + 1\) and \(x + 2\).
  2. Translate "add up to 72": \(x + (x + 1) + (x + 2) = 72\).
  3. Combine like terms: \(3x + 3 = 72\). Subtract 3: \(3x = 69\). Divide: \(x = 23\).
  4. Answer: the integers are 23, 24, and 25. Check: \(23 + 24 + 25 = 72\).

Example 2: a jacket is 30% off, and the sale price is $56. What was the original price?

  1. Name: let \(x\) = the original price in dollars.
  2. 30% off means you pay the original minus 30% of it: \(x - 0.30x = 56\).
  3. Combine: \(0.70x = 56\). \(1x - 0.30x = 0.70x\). You pay 70% of the price.
  4. Divide: \(x = 56 \div 0.70 = 80\). The original price was $80.
  5. Check: 30% of $80 is $24, and \(80 - 24 = 56\). It works.

Step 4 of 5Practice

Write "Let \(x\) = ..." on paper first. Type only the number that answers the question (no units), then press Check.

One number is 6 more than another. Their sum is 40. What is the smaller number?

Let \(x\) = the smaller number. The other is \(x + 6\).

\(x + (x + 6) = 40\), so \(2x + 6 = 40\), \(2x = 34\), \(x = 17\).

Check: \(17 + 23 = 40\). Answer: 17.

Seven less than three times a number is 20. What is the number?

Let \(x\) = the number. "Three times a number" is \(3x\), and "seven less than" that is \(3x - 7\).

\(3x - 7 = 20\), so \(3x = 27\) and \(x = 9\).

Check: \(3 \cdot 9 - 7 = 20\). Answer: 9.

The sum of two consecutive odd integers is 56. What is the larger one?

Let \(x\) = the smaller odd integer. The next odd integer is \(x + 2\).

\(x + (x + 2) = 56\), so \(2x + 2 = 56\), \(2x = 54\), \(x = 27\).

The question asks for the larger one: \(27 + 2 = 29\). Answer: 29.

A rectangle has a perimeter of 46 feet. Its length is 5 feet more than its width. What is the width, in feet?

Let \(w\) = the width. Then the length is \(w + 5\).

\(P = 2l + 2w\) gives \(2(w + 5) + 2w = 46\), so \(4w + 10 = 46\), \(4w = 36\), \(w = 9\).

Check: length 14, and \(2(14) + 2(9) = 46\). Answer: 9 feet.

A pair of shoes is 25% off. The sale price is $54. What was the original price, in dollars?

Let \(x\) = the original price. You pay 75% of it: \(x - 0.25x = 0.75x\).

\(0.75x = 54\), so \(x = 54 \div 0.75 = 72\).

Check: 25% of $72 is $18, and \(72 - 18 = 54\). Answer: $72.

Two cars leave the same town at the same time, driving in opposite directions. One goes 55 mph and the other goes 65 mph. After how many hours are they 300 miles apart?

Let \(t\) = the time in hours. Each car's distance is rate · time: \(55t\) and \(65t\).

Opposite directions, so the distances add: \(55t + 65t = 300\), so \(120t = 300\) and \(t = 2.5\).

Check: \(137.5 + 162.5 = 300\). Answer: 2.5 hours.

Sergio drives to a city at 60 mph and drives back home at 40 mph. The round trip takes 5 hours of driving. How many miles away is the city?

Let \(d\) = the distance one way. Time is distance ÷ rate, so going takes \(\dfrac{d}{60}\) hours and coming back takes \(\dfrac{d}{40}\) hours.

\(\dfrac{d}{60} + \dfrac{d}{40} = 5\). The LCD is 120: \(2d + 3d = 600\), so \(5d = 600\) and \(d = 120\).

Check: 2 hours there and 3 hours back is 5 hours. Answer: 120 miles.

A price goes up by 15%, and the new price is $92. Which equation finds the old price \(x\)?

  1. \(x + 0.15x = 92\)
  2. \(x + 0.15 = 92\)
  3. \(x - 0.15x = 92\)
  4. \(x + 15x = 92\)

The new price is the old price plus 15% of it: \(x + 0.15x = 92\). Answer (a).

(b) adds 15 cents, not 15%. (c) is a price going down. (d) uses 15 instead of 0.15.

Solving: \(1.15x = 92\), so the old price was \(x = 80\) dollars.

Step 5 of 5Wrap-up

You can now set up and solve word problems about numbers, consecutive integers, percent discounts, perimeter, and distance, and answer the question that was actually asked.

Read or watch next: OpenStax Intermediate Algebra 2e, 2.2 Use a Problem Solving Strategy (free textbook, with more practice at the end of the section).

Also good: Khan Academy: Pre-algebra: Equations word problems (short videos). Quick rules: Equations and inequalities cheat sheet.

Something unclear? Ask your teacher bot a follow-up question, for example: "How do I know which amount to call x?" It knows this course and can explain things another way.