Overview
Distinct parallel nonvertical lines share a slope. Perpendicular nonvertical lines have slopes whose product is −1, so one is the negative reciprocal of the other. Horizontal and vertical lines form a separate perpendicular pair. A supplied point determines the intercept after the slope is chosen.
Consider this example: What is the slope of a line parallel to y = −4x + 9? Parallel lines share their rate of change, so the slope is −4. The positive reciprocal 1/4 would instead be perpendicular.
The method also handles a different case: What is the slope of a line perpendicular to y = (2/3)x − 5? Take the reciprocal 3/2 and reverse its sign, giving −3/2. The product (2/3)(−3/2) = −1 verifies perpendicularity.
For a further application, consider: Which line is perpendicular to x = −4 and passes through (2, 7)? The original line is vertical. Its perpendicular is horizontal, so y stays constant at the supplied point’s y coordinate 7. Therefore the equation is y = 7.
Changing only the sign of a slope gives a perpendicular line only when its magnitude is 1. A reciprocal must also change sign. For a new relationship, decide whether slopes should be equal or multiply to −1. Then use the point to find the intercept and verify it in the resulting equation.