🎯 Learning Objectives
- How to translate every common word-problem phrase into algebra.
- The 4-step framework that solves any linear word problem.
- How to recognize when slope = rate and y-intercept = starting amount.
- How to interpret slope, intercepts, and constants in real-world contexts.
- Strategies that catch the trap answers.
- Practice questions with worked solutions.
1. Translation Dictionary
English Phrase | Math Symbol |
is, was, equals, will be | = |
sum / total / increased by / more than | + |
difference / less than / decreased by | − |
product of / times / of (with %) | × |
per / each / for every | ÷ |
a number / unknown quantity | x (or any letter) |
twice a number | 2x |
at least | ≥ |
at most | ≤ |
2. The 4-Step Word-Problem Framework
STEP 1: DEFINE: 'Let x = …' (write it down)
STEP 2: TRANSLATE every sentence into an equation
STEP 3: SOLVE the equation
STEP 4: RE-READ and answer the EXACT thing asked
3. Linear Models in Real Life
Most SAT word problems fit the template:
Generic linear model
y = (rate)·x + (starting amount)
Real-Life Quantity | Maps To |
Cost per item × number of items + flat fee | y = m·x + b |
Distance = speed × time + initial distance | d = r·t + d₀ |
Population = growth rate × years + starting population | P = k·t + P₀ |
Account balance = monthly addition × months + initial deposit | B = a·m + B₀ |
4. Interpreting Slope and Intercept in Context
💡 What Each Piece Means in a Word Problem
- Slope (m): the RATE of change. 'How much y changes for each unit increase in x.'
- y-intercept (b): the STARTING value. 'What y is when x = 0.'
- x-intercept: When does y reach zero? Often the 'breakeven' or 'all-used-up' moment.
📝 Example: A taxi charges $3 base fare plus $2 per mile. Write a model.
Define: Let C = total cost ($), m = number of miles.
Translate: C = 2m + 3
Slope = 2: $2 charge per additional mile.
y-int = 3: $3 base fare even before driving anywhere.
5. Strategies
⚡ STRATEGY 1: Identify the 'per' to find slope
In a linear model, the constant rate of change gives the slope. Check the dependent variable, input variable, and their units before using a stated rate.
⚡ STRATEGY 2: Identify the flat fee to find y-intercept
Look for words like 'starting', 'initial', 'base', 'flat', 'one-time'. That number is your y-intercept (b).
⚡ STRATEGY 3: RE-READ the question
Check whether the question asks for x or for an expression such as x + 5 or 2x. Report the requested quantity.
⚠️ Common Mistakes
- Mixing up which variable is which (writing one equation for x, another for y).
- Forgetting to convert units (minutes vs. hours, dollars vs. cents).
- Solving for the wrong variable. The question wants 'the number of seats sold', not 'x'.
- Treating 'per' as something other than a rate.
- Writing 'x − 5' for 'five less than x' (correct) but 'x − 5' for '5 less x' (also correct): but '5 less than 5x' = 5x − 5, NOT 5 − 5x.
6. Summary
📌 Key Takeaways
- Framework: Define → Translate → Solve → Re-read.
- Linear model: y = (rate)·x + (start)
- Slope: The rate of change ('per' something).
- y-intercept: The starting/initial amount.
- Trap: Always re-read what the question is actually asking.
7. Practice: 10 Questions
Q1. A gym charges a $50 sign-up fee plus $30 per month. Write a model for total cost C after m months.
A) C = 50m + 30 B) C = 30m + 50 C) C = 80m D) C = 30 + 50m
Hint: Per-rate is slope (30); flat fee is y-int (50).
Q2. In the model y = 4x + 12, what does 12 represent if x is hours and y is dollars?
A) Hourly rate B) Total earnings after 1 hour C) Starting amount D) Tax
Hint: y-int = starting value at x = 0.
Q3. A printer prints 25 pages per minute and starts with 100 pages already in the tray. After m minutes, total pages output is…
A) 100m + 25 B) 25m + 100 C) 25m D) 100 − 25m
Hint: 'Per minute' rate = 25 (slope); 'starts with' 100 (y-int).
Q4. A linear model: 'cost grows by $5 per pound, with a $2 packing fee.' Write the equation.
A) C = 2p + 5 B) C = 5p + 2 C) C = 5 + 2p D) C = 7p
Hint: Per-pound rate = 5 (slope); flat fee = 2 (y-int).
Q5. A car starts 60 miles from home and drives away at 50 mph. Distance from home after t hours is…
A) 50t + 60 B) 60t + 50 C) 50t − 60 D) 60 − 50t
Hint: Speed is rate (slope = 50); starting distance = 60.
Q6. The equation y = 7x − 15 models profit y after selling x items. The shop breaks even when…
A) x = 0 B) x ≈ 2.1 C) x = 7 D) x = 15
Hint: Break even when y = 0: x = 15/7 ≈ 2.1.
Q7. A water tank has 200 gallons and is being drained at 8 gallons per minute. Volume after m minutes is…
A) 8m + 200 B) 200 + 8m C) 200 − 8m D) 8m − 200
Hint: Drain → negative slope. Start at 200.
Q8. A phone plan: $20/month plus $0.10 per text. Total cost for one month if 100 texts sent?
A) $25 B) $30 C) $35 D) $40
Hint: 20 + 0.10(100) = 20 + 10 = $30.
Q9. A linear function f(x) satisfies f(0) = 7 and f(2) = 13. What is f(5)?
A) 16 B) 19 C) 22 D) 25
Hint: Slope = (13 − 7)/2 = 3. f(x) = 3x + 7. f(5) = 22.
Q10. The cost C in dollars to produce x widgets is C = 2x + 50. What is the cost per widget?
A) $2 B) $50 C) $52 D) $25
Hint: Per-unit cost = slope = 2.
Answer Key & Worked Solutions
# | Answer | Reasoning |
Q1. | B) C = 30m + 50 | Rate × variable + start. |
Q2. | C) Starting amount | y-int interpretation. |
Q3. | B) 25m + 100 | Rate of 25, starting 100. |
Q4. | B) C = 5p + 2 | Slope from 'per pound' = 5. |
Q5. | A) 50t + 60 | Speed × time + initial distance. |
Q6. | B) x ≈ 2.1 | Set y = 0 and solve. |
Q7. | C) 200 − 8m | Draining → minus the rate × time. |
Q8. | B) $30 | 20 + 0.10·100 = 30. |
Q9. | C) 22 | Slope 3, y-int 7. f(5) = 22. |
Q10. | A) $2 | Cost per item is the slope. |