Problem 54 - Introductory and Intermediate Algebra for College Students 4th

Question

What does it mean if the slope of a line is undefined?

Step-by-Step Solution

Answer

An undefined slope refers to a vertical line on the graph, where there is no horizontal change between any two points on the line. This leads the denominator ('run' or difference of 'x' coordinates) to be zero, making the slope undefined as division by zero is undefined in mathematics.

Step 1: Understanding the Concept of Slope

The slope of a line is a measure that indicates the steepness of the line. It's calculated by taking the difference of the 'y' coordinates (vertical change, also known as 'rise') and dividing it by the difference of the 'x' coordinates (horizontal change, also known as 'run'), in two different points on the line. Represented mathematically as \( \frac{ÿ_2 - y_1}{x_2 - x_1} \).

Step 2: Define an Undefined Slope

A slope is undefined when the denominator (the 'run' or the difference in 'x' coordinates) is zero. Division by zero is undefined in mathematics. This situation arises when the line is vertical, meaning it goes straight up and down, so there's no horizontal change. Consequently, the formula for slope becomes \( \frac{y_2 - y_1}{0} \), which is undefined.

Step 3: Visualizing an Undefined Slope

To further understand this, picture a vertical line on a graph. No matter which two points are chosen on this line, the 'x' coordinates will be the same because the line does not slant either to the right or left. Thus, the difference between these 'x' coordinates (x_2 - x_1) is zero, leading to an undefined slope.