Linear relationships are one of the most heavily tested topics on the SAT Math section, showing up in roughly a fifth of all questions across both modules. This category covers linear functions written in function notation like f(x), linear equations in the form y = mx + b, real-world rate problems involving constant growth or decline, and word problems that ask you to translate a situation into an algebraic model. The College Board tests these ideas in several ways: evaluating a function at a given input, finding the slope or y-intercept of a real-world scenario, building an equation from a table or a written description, and interpreting what a constant or coefficient means in context.
The 18 problems below give you a complete, representative sample of how linear relationships appear on the real exam, from straightforward function evaluation to multi-step word problems involving cost, growth, and depreciation. Every problem includes a full, step-by-step answer explanation so you can see exactly how to set up the equation, not just what the final answer is. Work through them in order, cover the explanation until you've committed to an answer, and pay close attention to problems that ask you to interpret a constant term or build an equation from a real-world description — these are the ones students miss most often on test day. When you're ready for more, our free SAT Math QBank has over 2,500 additional practice problems across every SAT Math topic, each with instant, detailed explanations.
For the function f(x) = (x + 8) / 5, what is the value of f(7) − f(−3)?
Substitute each input into the function separately. f(7) = (7 + 8)/5 = 15/5 = 3. f(−3) = (−3 + 8)/5 = 5/5 = 1.
Subtract: f(7) − f(−3) = 3 − 1 = 2. The correct answer is C.
A magazine debuted with 1,329 subscribers and gains 270 new subscribers every quarter. If the number of subscribers y after x quarters is written as y = ax + b, what is the value of b?
In the form y = ax + b, a is the rate of change per quarter and b is the starting value, when x = 0 (the moment the magazine debuted).
Since the magazine started with 1,329 subscribers before any quarterly growth was added, b = 1,329.
John earns $14.70 per hour. When he works z hours, he earns 14.70z dollars. Which expression gives his earnings if he works 5z hours?
Earnings equal the hourly rate times hours worked. His hourly rate stays $14.70, and the new number of hours is 5z (not 5 more hours — 5 times as many hours).
Substituting 5z in for the number of hours worked into the original earnings expression 14.70(hours) gives 14.70(5z), which is the same as 5(14.70z). The correct answer is D.
Cranjis rented a motorcycle for one day from a company that charges $100 per day plus $0.35 per mile driven. If she was charged a total of $149, for how many miles of driving was Cranjis charged?
Set up the total cost equation: 100 + 0.35m = 149, where m is the number of miles driven.
Subtract 100 from both sides: 0.35m = 49. Divide both sides by 0.35: m = 140. The correct answer is B.
The acceleration due to gravity on Jupiter is 7 ft/s² less than 3 times the acceleration due to gravity on Venus. If the acceleration on Jupiter is 80 ft/s², what is the acceleration, in ft/s², on Venus?
Translate the sentence into an equation. Let V be Venus's acceleration. "7 less than 3 times Venus" means Jupiter = 3V − 7.
So 3V − 7 = 80. Add 7 to both sides: 3V = 87. Divide by 3: V = 29 ft/s².
The equation y = 0.2x models the number of different exercises, y, a fitness model performs during an x-minute workout. How many exercises did she perform in a 40-minute session?
Substitute x = 40 into the equation: y = 0.2(40).
y = 8. The correct answer is C.
The profit, p, from producing and selling f frisbees is given by p = 17f − (6f + t), where t is a constant. If 300 frisbees are produced and sold for a profit of $2,150, what is the value of t?
Substitute f = 300 and p = 2,150 into the equation: 2,150 = 17(300) − (6(300) + t).
Simplify: 2,150 = 5,100 − (1,800 + t), so 2,150 = 3,300 − t. Solving for t: t = 3,300 − 2,150 = 1,150.
A company's total cost c(x), in euros, to produce x pantaloons is c(x) = mx + 360, where m is a constant. The cost to produce 50 pantaloons is €660. What is the total cost, in euros, to produce 500 pantaloons?
First find m using the known point. c(50) = 660, so 50m + 360 = 660. Subtract 360: 50m = 300, so m = 6.
Now find c(500): c(500) = 6(500) + 360 = 3,000 + 360 = 3,360.
For the function f(x) = (x + 2) / 3, what is the value of f(−2)?
Substitute x = −2 into the function: f(−2) = (−2 + 2) / 3.
f(−2) = 0 / 3 = 0. The correct answer is B.
T = 2,000 + 20a models Penny's total points, where a is the number of apples she caught and 2,000 is a starting bonus. If Penny scored 4,400 points, how many apples did she catch?
Set T equal to 4,400: 4,400 = 2,000 + 20a.
Subtract 2,000 from both sides: 2,400 = 20a. Divide by 20: a = 120. The correct answer is B.
The top floor of a trapezoidal building has 4 rooms. There are 45 floors total, and each floor has 3 more rooms than the floor above it. Which expression gives the total number of rooms on the bottom floor?
The top floor (floor 1) has 4 rooms, and each floor after it adds 3 more rooms. So the number of rooms on floor n follows the pattern 4 + 3(n − 1), since floor 1 gets zero increases, floor 2 gets one increase, and so on.
The bottom floor is floor 45, so the increases applied number 45 − 1 = 44. That gives 4 + 3(45 − 1). The correct answer is A.
Paige was 37 inches tall at age 4 and 41 inches tall at age 5. If her height increases by the same amount each year between ages 4 and 9, how many inches tall will she be at age 8?
Find the constant yearly growth rate: 41 − 37 = 4 inches per year.
From age 4 to age 8 is 4 years of growth, so add 4 × 4 = 16 inches to her starting height: 37 + 16 = 53 inches.
Using the growth factor table, what is the approximate age of a Norway maple tree with a diameter of 18 inches? (Norway maple growth factor: 3.0)
The problem states that age is found by multiplying diameter by the species' growth factor.
Age = 18 × 3.0 = 54 years. The correct answer is C.
A ponderosa pine (growth factor 2.3) and a hackberry (growth factor 3.1) each currently have a 10-inch diameter. What will be closest to the difference in their diameters 10 years from now?
Since age = diameter × growth factor, diameter = age ÷ growth factor. Over the next 10 years, each tree's age increases by 10, so each tree's diameter increases by 10 ÷ growth factor.
Pine's diameter increase: 10 ÷ 2.3 ≈ 4.35 inches. Hackberry's diameter increase: 10 ÷ 3.1 ≈ 3.23 inches. The difference is about 4.35 − 3.23 ≈ 1.1 inches. The correct answer is B.
A water company charges Franklin $0.04 per gallon. If he was charged $40, how many gallons of water did he use?
Set up the equation total cost = rate × gallons: 40 = 0.04g.
Divide both sides by 0.04: g = 1,000. The correct answer is D.
The function g is defined as g(x) = 4x/5 + 4. What is the value of g(−50)?
Substitute x = −50 into the function: g(−50) = 4(−50)/5 + 4.
4(−50) = −200, and −200/5 = −40. So g(−50) = −40 + 4 = −36. The correct answer is C.
A bicycle shop opened with 42 bikes. Each week for 3 months, 21 bikes were purchased and 14 were sold on average. Which equation models the inventory, b, at t weeks after opening?
Find the net weekly change in inventory: 21 bikes purchased minus 14 bikes sold = a net gain of 7 bikes per week.
Starting inventory is 42, so the model is b = 7t + 42, where 7 is the weekly rate of increase. The correct answer is C.
A company buys a 3D printer for $4,360. It depreciates at a constant rate for 8 years, after which it's worth nothing. How much is it worth 2 years after purchase?
Since the printer loses all $4,360 of value over 8 years at a constant rate, it depreciates by 4,360 ÷ 8 = $545 per year.
After 2 years, it has lost 2 × 545 = $1,090 in value: 4,360 − 1,090 = $3,270. The correct answer is C.
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