Identify lines parallel/perpendicular using slope relationships. | Linear equations in two variables | SAT Math

Parallel lines match slopes; perpendicular lines use negative reciprocal slopes.

核心知识

Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals: if one slope is $𝑚$ , the other is $- 1 𝑚$ .

深入理解

Two lines are parallel when they have the same steepness and direction — same slope, different y-intercepts. They never intersect. Two lines are perpendicular when they meet at a 90° angle, which happens when their slopes multiply to $- 1$ .

To check: convert both equations to slope-intercept form and compare slopes. If $𝑚 1 = 𝑚 2$ , the lines are parallel. If $𝑚 1 \cdot 𝑚 2 = - 1$ , they're perpendicular.

The negative reciprocal flips the fraction and changes the sign. The negative reciprocal of $23$ is $- 32$ . The negative reciprocal of $- 4$ (which is $- 41$ ) is $14$ .

分步讲解

Convert both equations to slope-intercept form to isolate the slopes.
Compare slopes: equal slopes → parallel; product of slopes = $- 1$ → perpendicular.
If asked to write an equation parallel or perpendicular to a given line, use the appropriate slope with a given point.
Plug into point-slope form and simplify.

常见误解

Taking the reciprocal but forgetting to negate — the perpendicular slope of $2$ is $- 12$ , not $12$ .
Assuming lines with the same y-intercept are parallel — parallel means same slope, not same intercept.
Thinking perpendicular slopes just have opposite signs — $3$ and $- 3$ are not perpendicular; $3$ and $- 13$ are.

题目

示例解析

Line $𝑝$ has the equation $𝑦 = 25 𝑥 + 3$ . Which of the following is the slope of a line perpendicular to line $𝑝$ ?

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