Math: Algebra, 3 original questions
Practice points on a line, equations through two points and perpendicular slopes in three original SAT-style questions with step-by-step answer explanations.
A two-variable linear equation describes a relationship between x and y, often a line. Substitute a known coordinate to find the other, or rewrite the equation as y = mx + b to read its slope and y-intercept. When given two points, calculate the change in y divided by the change in x, then use either point to find the intercept. Track coordinate order and signs. Check that a proposed equation fits every given point rather than only one.
Method at a glance
Use these steps in order, then apply them to the questions below.
Decide whether you need a coordinate, slope, intercept or entire line equation. Keep x and y in the stated order.
Substitute into Ax + By = C for a missing coordinate. Use y = mx + b when interpreting slope or intercept.
For two points, subtract coordinates in the same order: change in y divided by change in x. A vertical line needs separate treatment.
Check all supplied points. For nonvertical perpendicular lines with nonzero slopes, the slopes multiply to −1.
These are original SAT-style questions written by PeakSAT, not official College Board questions. Choose an answer and check it to see why each choice is right or wrong, or open the worked answer.
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The point (4, y) lies on 3x + 2y = 18. What is y?
Correct answer
Choice A: 3
Explanation
Substitute x = 4: 3 × 4 + 2y = 18. Subtract 12 to obtain 2y = 6, then divide by 2 to get y = 3. The point (4, 3) makes the left side 18.
Why each choice is right or wrong
Correct: 12 + 2 × 3 = 18.
6 is the value of 2y, not y.
This divides 18 by 2 without subtracting the x-term.
12 is the x-term 3 × 4, not the y-coordinate.
Which equation describes the line through (2, 7) and (6, 15)?
Correct answer
Choice C: y = 2x + 3
Explanation
The slope is (15 − 7)/(6 − 2) = 8/4 = 2. Use (2, 7) in y = 2x + b: 7 = 4 + b, so b = 3. The equation y = 2x + 3 gives 7 at x = 2 and 15 at x = 6.
Why each choice is right or wrong
The slope is correct, but this equation gives 11 rather than 7 at x = 2.
It fits the first point but gives 11, not 15, at x = 6.
Correct: it fits both points and has the required slope 2.
It fits the first point but has slope 4 and gives 23 at x = 6.
A line is perpendicular to y = (3/4)x + 2. What is its slope?
Correct answer
Choice D: −4/3
Explanation
The given slope is 3/4. A perpendicular nonvertical line has the negative reciprocal, −4/3, because (3/4)(−4/3) = −1. The intercept 2 does not affect the direction of the line.
Why each choice is right or wrong
This changes the sign but does not take the reciprocal.
This is the slope of a parallel line.
This takes the reciprocal but omits the negative sign.
Correct: the negative reciprocal is −4/3.
Reversing only one subtraction in a slope calculation. Subtract both coordinates in the same point order.
Treating the constant in standard form as the y-intercept. Set x = 0 or solve for y first.
Changing only the sign for a perpendicular slope. Use the negative reciprocal when both slopes are defined and nonzero.
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Yes. A line contains many coordinate pairs. A single known coordinate or another independent condition can narrow the possibilities.
Yes. Substitute x = 0 or read b after writing a nonvertical line as y = mx + b.
It is undefined because the change in x is zero. A horizontal line, with slope zero, is perpendicular to a vertical line.
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