SAT Math · Algebra

Linear Functions

Only 12% correct on real student attempts. f(x) = mx + b. Three moves cover this entire skill.

The rule

Linear functions have constant slope: the change in output per 1-unit increase in input. The form is f(x) = mx + b. Build slope from two points using (y₂ − y₁)/(x₂ − x₁). When an equation is "true for all x," expand, group like terms, then match x-coefficients and constants across the equals sign.

Worked example

The equation shown gives the estimated amount of gas g, in gallons, that remains in the gas tank of a car after being driven x miles, where 0≤x≤3770 \leq x \leq 377. What is the estimated amount of gas, in gallons, that remains in the gas tank of the car when x = 290?

g=13−x29g = 13 - \frac{x}{29}

  • A.

    13

  • B.

    0

  • C.

    3

  • D.

    16

Walk through it

C — Using the given formula, substitute x=290x = 290 to solve for gg; the answer is 3.

The traps

Wrong substitution.

The SAT gives f(expression) and a number. A trap answer comes from misapplied parentheses, a sign flip, or solving f(x) = number directly for x instead of for the expression inside. Always write out the substitution with parentheses before simplifying.

Reversed slope or wrong point.

A trap inverts the slope fraction, or calculates b from one point but checks with none. Confirm your equation works for both given points.

Individual pieces as answers.

In "true for all x," the answer choices include individual coefficients or intermediate values instead of the requested sum. Combine fully, then read off only what the question asks for.

The procedure

  1. Find slope from two points: subtract y's, subtract x's, divide in that order.
  2. Plug slope and one point into f(x) = mx + b to get b. Verify with the other point.
  3. For "all x" problems: expand, collect x-terms and constants, set coefficients equal, solve. Answer exactly what is asked — often a sum, not an individual letter.
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Drill this skill

Rule, traps, and procedure drawn from the verified skill playbook for “Linear Functions” (drafted against real questions, verified by a second independent pass — see lib/content/playbooks/linear-functions.md). Worked example is a real, verified SAT practice question from the live question bank, not generated for this lesson.