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Chapter 3: Problem 54

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

Short Answer

Expert verified
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 by step solution

01

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 \( \frac{每_2 - y_1}{x_2 - x_1} \).
02

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.
03

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.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. Remember that a solution of an equation in two variables is an ordered pair. Let \(x=0\) and find a solution of \(x+2 y=0\)

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((-2,2)\) and parallel to the line whose equation is \(2 x-3 y=7\)

Simplify: \(\quad 3\left(12 \div 2^{2}-3\right)^{2}\). (Section \(1.8,\) Example 6 )

Use a graphing utility to graph each equation. You will need to solve the equation for \(y\) before entering it. Use the graph displayed on the screen to identify the \(x\) -intercept and the \(y\) -intercept. $$3 x-y=9$$

Solve for \(h: \quad V=\frac{1}{3} A h .\) (Section 2.4, Example 4)
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