Problem 56 - Introductory Algebra for College Students

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. That is, for two perpendicular lines with slopes \( m_1 \) and \( m_2 \), the equation \( m_1 \cdot m_2 = -1 \) holds true.

Step 1: Understanding Slope

Slope of a line is a measure of the degree to which the line tilts up or down. Mathematically, it is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. It is usually represented by the letter 'm'.

Step 2: Defining Perpendicular Lines

Perpendicular lines are two lines that intersect at a right angle (90 degrees).

Step 3: Relationship Between Slopes

If two lines are perpendicular, the product of their slopes is -1. In other words, if line 1 has a slope \( m_1 \) and line 2 has a slope \( m_2 \), and if the lines are perpendicular, then \( m_1 \cdot m_2 = -1 \). This means the slope of one line is the negative reciprocal of the slope of the other line.

Step 4: Recap

So to summarize, if two lines are perpendicular, their slopes are negative reciprocals of each other. This is because the product of the slopes of two perpendicular lines is -1.