Translating Word Problems into Linear Equations
By the end of this lesson, you’ll be able to:
- Translate verbal descriptions into algebraic expressions and equations.
- Identify and define variables clearly.
- Build and solve one-variable linear equations from real-world scenarios.
- Recognize common structures in linear word problems.
Key Ideas
Translating a word problem means taking a sentence in English and turning it into math.
The workflow stays the same across most problems:
Define the variable — decide exactly what \(x\) represents.
Translate common words:
- “total,” “sum,” “altogether” → addition
- “difference,” “less than,” “fewer than” → subtraction
- “times,” “product,” “per,” “each,” “rate” → multiplication
- “is,” “equals,” “results in” → equals sign
- “total,” “sum,” “altogether” → addition
Build the equation — match the structure of the statement.
Solve and interpret — check that your answer makes sense in context.
Words like “less than” reverse the order.
For example, “5 less than a number” = \(x - 5\), not \(5 - x\).
Common Problem Types
Cost / Fee Structure
A typical form is:
\[ \text{Total Cost} = \text{Fixed Fee} + (\text{Rate})(\text{Units}) \]
Distance / Speed / Time
Use:
\[ d = rt \]
Age Relationships
Ages relate by addition or subtraction, often resulting in a sum or comparison.
“Sum of Parts”
Two or more quantities combine to make a total.
Comparisons & Translations
These often hinge on phrases like “less than,” “more than,” or “twice as much.”
Strategies
- Always define the variable first — translation becomes much easier.
- Sketch the relationship (a small table or diagram) when the wording is tricky.
- Keep operation words in mind — they guide the structure of the equation.
- Rewrite the sentence with blanks (e.g., “cost = fee + rate × ___”).
- If a phrase feels ambiguous, try plugging in a simple number to test direction.
- After solving, check units (dollars, miles, years, etc.) to confirm your answer makes sense.
Worked Examples
Example 1 — Cost Problem
A gym charges a $25 sign-up fee plus $40 per month. Someone spent $185. How many months were they a member?
Let \(x\) = number of months.
Equation:
\[ 25 + 40x = 185 \]
Solve:
\[ 40x = 160 \]
\[ x = 4 \]
Example 2 — Distance / Speed Problem
A car travels at 55 mph. How long does it take to travel 165 miles?
Let \(t\) = time in hours.
\[ 55t = 165 \]
\[ t = 3 \]
Example 3 — Age Problem
Ben is \(b\) years old. Maria is 8 years older. Their ages sum to 40. Find Ben’s age.
Let \(b\) = Ben’s age.
Maria = \(b + 8\).
Equation:
\[ b + (b + 8) = 40 \]
\[ 2b + 8 = 40 \]
\[ b = 16 \]
- Translating “less than” incorrectly (order matters).
- Forgetting to include fixed fees before rates.
- Mixing units like hours/minutes.
- Solving without defining the variable first.
Practice Problems
- A meal costs $30 plus $5 per drink. The total was $65. How many drinks?
- A taxi charges $3.50 plus $1.20 per mile. The total was $21.50. How many miles?
- Two siblings’ ages add to 33. One is twice the other’s age. Find both ages.
- A number increased by 4 equals 52. What is the number?
- You earn $9 per hour. How many hours did you work if you earned $225?
1.
Let \(x\) = drinks
\[
30 + 5x = 65
\]
\[
x = 7
\]
2.
Let \(m\) = miles
\[
3.5 + 1.2m = 21.5
\]
\[
m = 15
\]
3.
Let \(s\) = younger sibling
Older = \(2s\)
\[
s + 2s = 33
\]
\[
s = 11,\quad \text{older} = 22
\]
4.
\[
x + 4 = 52
\]
\[
x = 48
\]
5.
\[
9x = 225
\]
\[
x = 25
\]
Summary
- Define your variable first—everything flows from that step.
- Translate key words into math operations.
- Build the equation that matches the real-world relationship.
- Solve and check that your answer makes sense in context.
- Replace numbers with \(x\) in the sentence to see the structure.
- Draw a mini table for cost, rate, time, or ages to organize info.
- Watch out for reversed phrasing like “___ less than ___.”
- After solving, ask: “Does this value make sense in real-world terms?”