Slope, Parallel & Perpendicular Lines

Overview

Slope measures the steepness of a line, and it decides how two lines relate. Parallel lines have equal slopes, and perpendicular lines have slopes that are opposite reciprocals — flip the fraction and change the sign, so the two slopes multiply to -1. This lesson computes slopes from two points and from equations, finds the slope of any line parallel or perpendicular to a given line, classifies pairs of lines, and finishes by writing the equation of a line through two points in slope-intercept form.

Vocabulary

Slope: The ratio of the vertical change (rise) to the horizontal change (run) of a line: m = (y2 - y1)/(x2 - x1) for two points (x1, y1) and (x2, y2) on the line. A line rising to the right has positive slope, a line falling to the right has negative slope, a horizontal line has slope 0, and a vertical line has undefined slope.
Slope-Intercept Form: The equation of a line written y = mx + b, where m is the slope of the line and b is its y-intercept.
y-Intercept: The y-coordinate of the point where a line intersects the y-axis; it is the value b in y = mx + b.
Parallel Lines (slope condition): Two nonvertical lines are parallel if and only if their slopes are equal.
Perpendicular Lines (slope condition): Two lines are perpendicular if and only if their slopes are opposite reciprocals — the product of their slopes is -1.
Opposite Reciprocals: Two numbers whose product is -1: to form the opposite reciprocal of a fraction, flip the fraction and change its sign. For example, the opposite reciprocal of 2/5 is -5/2.

Main Ideas

Slope is computed the same way every time: m = (y2 - y1)/(x2 - x1). The order of the points does not matter, as long as the subtraction order matches in the numerator and the denominator.
To read the slope from an equation, get the equation into slope-intercept form y = mx + b; the coefficient of x is the slope.
Parallel lines have equal slopes — a line parallel to a given line keeps the given slope exactly.
Perpendicular lines have opposite reciprocal slopes: flip the fraction and change the sign. The product of the two slopes is -1.
To classify two lines, compute both slopes: equal slopes mean parallel, opposite reciprocals mean perpendicular, and anything else means the lines intersect without being perpendicular.
To write an equation through two points: find the slope, substitute the slope and one point into y = mx + b to solve for b, then write the finished equation.