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Math · Checkpoint 15 of 30 · Algebra · ~15 min
Linear equations & functions
Slope, intercepts, and word problems — the single most-tested skill family on SAT Math.
The 30-second version
a line is y = mx + b: m is the rate of change (slope), b is the starting value (y-intercept). Every linear word problem on the SAT is asking you to identify one of those two things or to solve for where two of them meet. If you can read "the plumber charges $40 plus $25 per hour" as y = 25x + 40 instantly, a huge slice of SAT Math becomes routine.
Why this matters
Algebra is the largest Math domain, and linear equations/functions are its heart. They also hide inside later topics — systems, inequalities, and even data questions about lines of best fit. This is the single highest-value checkpoint in the Math phase.
Learn it
Solving: keep the equation balanced
3(x − 2) = 2x + 5 → distribute: 3x − 6 = 2x + 5 → subtract 2x: x − 6 = 5 → x = 11. Distribute carefully, collect like terms, undo operations in reverse order. Most "hard" linear equations are easy equations with one fraction or one set of parentheses added.
Slope: rise over run, and what it means
Through points (x₁, y₁) and (x₂, y₂): m = (y₂ − y₁) / (x₂ − x₁). In context, slope is always a rate with units — dollars per hour, meters per second, students per year. When a question asks "what does the 25 mean?", the answer is a rate sentence: the charge increases by $25 for each additional hour.
Intercepts: where lines touch the axes
y-intercept: set x = 0 — the starting value (the plumber's $40 base fee). x-intercept: set y = 0 — where the output runs out (when a draining tank hits empty). "Interpret the intercept" questions are asking for these sentences.
Word problems: translate, then solve
Circle the rate (per, each, every → slope) and the flat amount (fee, initial, starts at → intercept). "A gym charges a $30 sign-up fee and $18 per month; when does the total reach $174?" → 18x + 30 = 174 → x = 8 months.
Common traps
- Answering x when the question asked 2x + 1. The SAT loves solving-for-an-expression; reread the final question line before answering.
- Sign slips distributing negatives: −2(x − 3) is −2x + 6.
- Swapping slope and intercept in context. The per-unit number is never the starting value.
- Parallel = same slope; perpendicular = negative reciprocal. Memorize both.
Try it
A phone plan costs $12 per month plus $0.05 per text. The function C(t) = 0.05t + 12 gives the monthly cost. What does the 12 represent?
Set t = 0: C(0) = 12. The 12 is the y-intercept — the cost with zero texts, i.e. the base monthly charge. The per-text cost is the slope, 0.05. Interpretation questions always come down to "is this number the rate or the start?"
Free resources for this checkpoint
Khan Academy Official Digital SAT Prep (opens in a new tab)
Free structured lessons, video instruction, and practice exercises for both SAT sections, built with College Board as the official practice partner. After a Bluebook practice test, you can link your results to get personalized practice recommendations.
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Practice next
Question Bank filter: Math → Algebra → Linear equations in one/two variables and Linear functions. Do these by hand first; the Desmos checkpoint will show you the graphing shortcut later.
Practice these skillsOriginal Digital SAT-style questions on what this checkpoint covers.