Parallel lines share the same slope, while perpendicular lines have slopes that are negative reciprocals of each other, always multiplying to negative 1. Learn to identify and combine both relationships, with a worked example verifying parallel and perpendicular slopes together.
Combining parallel and perpendicular lines
Parallel lines share the same slope. Perpendicular lines have slopes that are negative reciprocals of each other — multiply the two slopes together and the answer is always −1.
Worked example
y = 2x + 1 and y = 2x − 3 both have slope 2, so they are parallel. The line y = −0.5x + 2 has slope −0.5, the negative reciprocal of 2 (since 2 × (−0.5) = −1), so it is perpendicular to both parallel lines.
Two parallel lines (slope 2) and a perpendicular line (slope −0.5), verified by 2 × (−0.5) = −1.
Solving combined problems
Problems that mix parallel and perpendicular conditions usually give you one line's equation and ask for another that is either parallel or perpendicular to it, often through a specific point. Start from the parallel line equation rules if the slopes should match, or take the negative reciprocal if the lines should meet at a right angle, then use a point and a slope to finish writing the equation, drawing on the same equation-solving skills used throughout linear equations.