SAT Math – Algebra
Linear Relationship Word Problems
Understanding how two variables change together in real-world contexts
Linear relationship word problems are a cornerstone of the SAT Math section, testing your ability to model real-world scenarios using linear functions. Unlike basic equation-solving, these questions ask you to understand how two quantities change together and what those changes mean in context.
These problems require you to interpret slope as a rate of change, understand y-intercepts as starting values, and translate between verbal descriptions, equations, and graphs. Mastering this skill is essential—linear relationship problems appear on virtually every SAT Math section and form the foundation for more advanced topics like systems of equations and data analysis.
What Are Linear Relationships?
Core Definition
A linear relationship is a relationship between two variables that creates a straight line when graphed. In these relationships, one variable changes at a constant rate relative to the other variable.
Where:
- \(m\) = slope (rate of change)
- \(b\) = y-intercept (starting or initial value)
- \(x\) = independent variable (input)
- \(y\) = dependent variable (output)
Real-World Linear Relationships
- Phone bills: Monthly charge = Fixed fee + (Rate per minute × Minutes used)
- Taxi fares: Total cost = Base fare + (Rate per mile × Distance)
- Salaries: Total pay = Base salary + (Commission rate × Sales)
- Water tanks: Current volume = Starting volume + (Fill rate × Time)
- Car rentals: Total cost = Daily rate × Days + Insurance fee
What the SAT Tests
- Writing linear functions to model scenarios
- Interpreting slope and y-intercept in context
- Evaluating functions at specific values
- Identifying which value in an equation represents a specific quantity
- Comparing rates of change between different scenarios
Essential Concepts & Formulas
Slope-Intercept Form
\(y = mx + b\)
\(m\) is the slope – the rate of change (how much \(y\) changes for every 1-unit increase in \(x\))
\(b\) is the y-intercept – the value of \(y\) when \(x = 0\) (starting value, initial amount, fixed fee)
Calculating Slope from Two Points
\(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{change in } y}{\text{change in } x} = \frac{\text{rise}}{\text{run}}\)
Slope represents the rate of change: for every 1 unit increase in \(x\), \(y\) changes by \(m\) units.
Point-Slope Form
\(y - y_1 = m(x - x_1)\)
Useful when you know the slope and one point \((x_1, y_1)\) on the line.
Understanding Slope and Intercept in Context
Slope Context Clues:
Words like "per", "each", "rate", "every" indicate the slope.
Example: "$5 per hour" → slope = 5
Y-Intercept Context Clues:
Words like "initial", "starting", "one-time", "fixed", "base" indicate the y-intercept.
Example: "$20 starting fee" → y-intercept = 20
Common Pitfalls & Expert Tips
❌ Confusing slope and y-intercept
The most common mistake! The y-intercept is the starting value (when \(x = 0\)), and the slope is the rate of change. Don't mix them up when translating word problems.
❌ Misidentifying which variable is independent vs. dependent
The independent variable (\(x\)) is what you control or changes naturally (time, quantity). The dependent variable (\(y\)) is what you measure or calculate (cost, distance).
❌ Forgetting units in context questions
If asked "What does the slope represent?", include units! Don't just say "5" – say "5 dollars per hour" or whatever the context requires.
❌ Plugging in values incorrectly
When evaluating \(y = 3x + 5\) at \(x = 4\), substitute carefully: \(y = 3(4) + 5 = 17\). Don't forget the multiplication!
✓ Expert Tip: Look for "per" and "starting" keywords
These keywords immediately tell you which number is the slope (per) and which is the y-intercept (starting). This makes writing equations much faster.
✓ Expert Tip: Always check if your answer makes sense
If someone works 0 hours, they shouldn't earn a commission (only base pay). If a tank starts empty, the y-intercept should be 0. Use logic!
✓ Expert Tip: Draw a quick mental picture
Visualize the scenario. Does the value increase over time? That's a positive slope. Start at zero? Y-intercept is 0. Mental models prevent errors.
Fully Worked SAT-Style Examples
A plumber charges a $50 service fee plus $30 per hour of work. Write a linear function \(C(h)\) that represents the total cost in dollars for \(h\) hours of work.
Solution:
Step 1: Identify the components
• Starting value (y-intercept): $50 service fee (paid regardless of hours)
• Rate of change (slope): $30 per hour
• Independent variable: \(h\) = hours of work
• Dependent variable: \(C\) = total cost
Step 2: Use the slope-intercept form
General form: \(y = mx + b\)
In this context: \(C = 30h + 50\)
Verification:
For 0 hours: \(C(0) = 30(0) + 50 = 50\) (just the service fee) ✓
For 2 hours: \(C(2) = 30(2) + 50 = 110\) (service fee + 2 hours work) ✓
Answer: \(C(h) = 30h + 50\)
The function \(f(x) = 4x - 7\) represents the relationship between \(x\) and \(f(x)\). What is the value of \(f(5)\)?
Solution:
Step 1: Understand what \(f(5)\) means
\(f(5)\) means "evaluate the function when \(x = 5\)"
Step 2: Substitute \(x = 5\) into the function
\(f(5) = 4(5) - 7\)
\(f(5) = 20 - 7\)
\(f(5) = 13\)
Answer: \(f(5) = 13\)
A water tank contains 500 gallons of water. Water is being added to the tank at a rate of 25 gallons per minute. The function \(W(t) = 25t + 500\) represents the amount of water \(W\) in gallons after \(t\) minutes. What does the 25 represent in this equation?
Solution:
Step 1: Identify the form of the equation
The equation \(W(t) = 25t + 500\) is in slope-intercept form: \(y = mx + b\)
Here, \(m = 25\) (coefficient of \(t\)) and \(b = 500\)
Step 2: Interpret the slope in context
The coefficient 25 is the slope, which represents the rate of change.
Since the problem states water is added at 25 gallons per minute, the slope represents this rate.
Understanding:
For every 1 minute that passes, the amount of water increases by 25 gallons.
The 500 represents the initial amount of water (when \(t = 0\)).
Answer: The 25 represents the rate at which water is added to the tank, which is 25 gallons per minute.
A cell phone plan costs $15 per month plus $0.10 per text message sent. The function \(C(t) = 0.10t + 15\) gives the monthly cost \(C\) in dollars when \(t\) text messages are sent. What does the 15 represent in this function?
Solution:
Step 1: Identify the form
\(C(t) = 0.10t + 15\) is in the form \(y = mx + b\)
The slope is \(m = 0.10\) and the y-intercept is \(b = 15\)
Step 2: Interpret the y-intercept
The y-intercept (15) is the value of \(C\) when \(t = 0\)
When \(t = 0\): \(C(0) = 0.10(0) + 15 = 15\)
This means even if you send 0 text messages, you still pay $15.
Context:
The 15 represents the fixed monthly cost (base plan fee) regardless of usage.
The 0.10 represents the variable cost per text message.
Answer: The 15 represents the fixed monthly cost of the cell phone plan in dollars.
A taxi service charges a fare that increases linearly with distance. For a 3-mile trip, the fare is $12. For a 7-mile trip, the fare is $20. What is the rate per mile?
Solution:
Step 1: Identify the two points
Point 1: (3 miles, $12) → \((x_1, y_1) = (3, 12)\)
Point 2: (7 miles, $20) → \((x_2, y_2) = (7, 20)\)
Step 2: Use the slope formula
\(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{20 - 12}{7 - 3} = \frac{8}{4} = 2\)
Interpretation:
The slope is 2, which means the fare increases by $2 for each additional mile.
Verification: From 3 miles to 7 miles (4 additional miles), the cost increases $8 (from $12 to $20). $8 ÷ 4 miles = $2 per mile ✓
Answer: The rate is $2 per mile.
A gym membership costs $200 to join and then $35 per month. Write an equation for the total cost \(C\) in dollars after \(m\) months of membership. Then, find the total cost after 8 months.
Solution:
Step 1: Identify the components
• One-time joining fee: $200 (this is the y-intercept, \(b = 200\))
• Monthly cost: $35 per month (this is the slope, \(m = 35\))
• Independent variable: \(m\) = number of months
• Dependent variable: \(C\) = total cost
Step 2: Write the equation
Using \(y = mx + b\) form:
\(C = 35m + 200\)
Step 3: Find the cost after 8 months
Substitute \(m = 8\):
\(C = 35(8) + 200\)
\(C = 280 + 200\)
\(C = 480\)
Verification:
Joining fee: $200
8 months at $35/month: \(8 \times 35 = 280\)
Total: \(200 + 280 = 480\) ✓
Answer: Equation: \(C = 35m + 200\)
Total cost after 8 months: $480
Sarah earns $12 per hour at her job with no additional pay. Write a function \(E(h)\) that represents her earnings in dollars after working \(h\) hours. How much does she earn after working 6.5 hours?
Solution:
Step 1: Analyze the situation
• Rate: $12 per hour (slope = 12)
• Starting amount: $0 (no sign-on bonus, no base pay)
• Since there's no starting amount, the y-intercept \(b = 0\)
Step 2: Write the function
Using \(y = mx + b\) with \(m = 12\) and \(b = 0\):
\(E(h) = 12h + 0\) or simply \(E(h) = 12h\)
Note: When the y-intercept is 0, we often omit it from the equation.
Step 3: Calculate earnings for 6.5 hours
\(E(6.5) = 12(6.5) = 78\)
Understanding:
When there's no initial fee or starting value, the relationship passes through the origin (0, 0).
This is called a proportional relationship: earnings are directly proportional to hours worked.
Answer: \(E(h) = 12h\)
Earnings after 6.5 hours: $78
A car rental company charges a daily rate plus a one-time processing fee. The total cost for renting a car for 4 days is $180. The daily rate is $40. Write an equation for the total cost \(C\) after \(d\) days.
Solution:
Step 1: Identify what we know
• Slope (daily rate): \(m = 40\)
• One point: When \(d = 4\), \(C = 180\) → Point (4, 180)
• Unknown: The y-intercept \(b\) (processing fee)
Step 2: Use the slope-intercept form
General form: \(C = md + b\)
With \(m = 40\): \(C = 40d + b\)
Step 3: Substitute the known point to find \(b\)
Substitute \(d = 4\) and \(C = 180\):
\(180 = 40(4) + b\)
\(180 = 160 + b\)
\(b = 20\)
Step 4: Write the complete equation
\(C = 40d + 20\)
Verification:
For 4 days: \(C = 40(4) + 20 = 160 + 20 = 180\) ✓
Interpretation: The processing fee is $20, and the daily rate is $40.
Answer: \(C = 40d + 20\)
A candle starts at 8 inches tall and burns at a rate of 0.5 inches per hour. Write a function \(H(t)\) that gives the height of the candle in inches after \(t\) hours. What does the slope represent?
Solution:
Step 1: Identify the components
• Starting height: 8 inches (y-intercept, \(b = 8\))
• Rate of change: The candle decreases by 0.5 inches per hour
• Since the height decreases, the slope is negative: \(m = -0.5\)
Step 2: Write the function
\(H(t) = -0.5t + 8\)
Note: We can also write this as \(H(t) = 8 - 0.5t\)
Step 3: Interpret the slope
The slope of \(-0.5\) means:
• For every hour that passes, the height decreases by 0.5 inches
• Negative slopes indicate a decreasing relationship
Verification:
At \(t = 0\): \(H(0) = -0.5(0) + 8 = 8\) inches (starting height) ✓
At \(t = 2\): \(H(2) = -0.5(2) + 8 = 7\) inches (burned 1 inch in 2 hours) ✓
At \(t = 4\): \(H(4) = -0.5(4) + 8 = 6\) inches ✓
Answer: \(H(t) = -0.5t + 8\)
The slope represents the rate at which the candle burns, which is 0.5 inches per hour (decreasing).
A streaming service had 5,000 subscribers in January and gains 250 new subscribers each month. Write a function \(S(m)\) for the number of subscribers after \(m\) months. In which month will the service reach 10,000 subscribers?
Solution:
Step 1: Write the function
• Starting subscribers (January): 5,000 → \(b = 5000\)
• Growth rate: 250 subscribers per month → \(m = 250\)
\(S(m) = 250m + 5000\)
Step 2: Set up equation to find when subscribers reach 10,000
We want \(S(m) = 10000\):
\(250m + 5000 = 10000\)
Step 3: Solve for \(m\)
\(250m = 5000\)
\(m = \frac{5000}{250} = 20\)
Step 4: Interpret the answer
\(m = 20\) means 20 months after January
Counting: January is month 0, so 20 months later is September of the following year
Verification:
\(S(20) = 250(20) + 5000 = 5000 + 5000 = 10000\) ✓
Answer: Function: \(S(m) = 250m + 5000\)
The service will reach 10,000 subscribers after 20 months (September of the next year).
Quick Reference: Slope vs. Y-Intercept
SAT Success Strategies
1. Identify Keywords First
Before writing any equation, circle words like "per," "each," "starting," and "initial." These instantly tell you what the slope and y-intercept are.
2. Always Include Units in Interpretation Questions
If the SAT asks "What does the 35 represent?" don't just say "the slope." Say "the rate of 35 dollars per month" or whatever the context requires.
3. Check Your Answer for Reasonableness
If someone works 0 hours, should they earn $0 or have a base salary? Does the value increase or decrease over time? Logic catches errors.
4. Practice Writing Equations from Scratch
Don't just solve given equations—practice creating equations from word problems. This is the harder skill and what the SAT tests most often.
5. Master Both Forms: \(y = mx + b\) and \(C = at + b\)
The SAT uses different variable names depending on context. Be comfortable with any letters—the structure is always the same.
Linear Relationships: The Foundation of SAT Algebra
Linear relationship word problems test your ability to model the world mathematically—a skill that extends far beyond standardized tests. Every time you calculate costs, predict outcomes, or analyze trends, you're using linear thinking. Master these concepts, and you'll not only excel on the SAT but also develop quantitative reasoning skills that serve you throughout college and your career. Focus on understanding the meaning behind the math, and the numbers will follow naturally.
