Read what a line's variables mean, then choose a form that exposes the requested feature: slope, intercept, or a point on the line.
A linear equation in two variables describes a line. The useful form depends on the task: y = mx + b exposes slope and intercept, while Ax + By = C can make substitution or a proportional comparison convenient.
Worked example: interpret the quantities
The examples below are original teaching exercises, not released SAT questions. A delivery service charges C dollars for a trip of d kilometres according to C = 2.5d + 8. The slope is $2.50 per kilometre, and the intercept is a fixed $8 charge.
For a 12-kilometre trip, C = 2.5(12) + 8 = 38. Doubling the distance does not double the total charge because the fixed charge is still added only once. The equation's units help distinguish a starting value from a rate.
Worked example: convert standard form
For 3x + 2y = 12, isolate y: 2y = −3x + 12, so y = −1.5x + 6. The slope is −3/2 and the y-intercept is 6. To find the x-intercept, set y = 0; then x = 4. The line passes through (0, 6) and (4, 0).
Check a proposed point by substituting both coordinates. The point (2, 3) works because 3(2) + 2(3) = 12; the point (3, 2) does not.
Worked example: parallel and perpendicular
A line parallel to y = −2x + 5 has slope −2. If it passes through (3, 4), use y = −2x + b and substitute the point: 4 = −6 + b, so y = −2x + 10.
A line perpendicular to it has slope 1/2 because (−2)(1/2) = −1. Horizontal and vertical lines require a separate check: y = 4 is horizontal, x = 3 is vertical, and they are perpendicular although the vertical line has no finite slope.
Read tables and graphs carefully
Compute slope as change in y divided by change in x using the same two points. An axis labelled in hundreds changes the physical units represented by a coordinate. Do not assume the visual steepness alone gives a numerical slope when the axes use different scales.
For a contextual line, interpret only the domain the prompt describes. A model for delivery distance need not make sense for negative distance, even though its algebraic line extends there.
Check the requested feature
A question about an intercept does not require solving for an arbitrary point. A question about the effect of increasing x by 4 can often be answered as 4m. Keep the requested variable and unit attached to the answer.
For two equations that must hold together, continue to systems of linear equations. Review sign, coordinate order, slope units, and form-conversion errors separately.
Official sources
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