Problem 56 - Introductory and Intermediate Algebra for College Students 4th

Question

If two lines are perpendicular, describe the relationship between their slopes.

Step-by-Step Solution

Answer

The slopes of two perpendicular lines are negative reciprocals of each other, i.e., the product of their slopes is -1. If the slope of one of the lines is 0, the other line (being vertical) has an undefined slope.

Step 1: Definition of Perpendicular Lines

Perpendicular lines are two lines that intersect at a right angle (90 degrees). In a Cartesian Plane, this can be visualized as two lines where if one line is sloping upwards, the other line is sloping downwards.

Step 2: Understanding Slope

The slope of a line in a coordinate plane is obtained using the formula \( m = \Delta y / \Delta x \); where \( \Delta y \) is the change in y-coordinates and \( \Delta x \) is the change in x-coordinates. It basically tells us about the steepness of the line.

Step 3: Relating Slopes of Perpendicular Lines

For two lines to be perpendicular, the product of their slopes should be -1. In other words, if one line has slope m1 and the other line has slope m2, then \( m1 x m2 = -1 \). If the slope of one of the lines is 0 (making it horizontal), then the other line, being vertical, has an undefined slope.