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Chapter 3: Problem 54

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

Short Answer

Expert verified
The slope of a line is undefined when the line is vertical in a graph, representing an infinite rate of change. This happens because the denominator in the slope formula becomes zero due to no horizontal change, and mathematically, division by zero is undefined.

Step by step solution

01

Definition of a slope

The slope of a line in a coordinate plane is defined as the ratio of the vertical change (often referred to as the 'rise') to the horizontal change (often referred to as the 'run') between any two points on the line. Mathematically, if the coordinates for two points on a line are \((x_1, y_1)\) and \((x_2, y_2)\), then the slope, represented by 'm', is given by: \(m = (y_2 - y_1) / (x_2 - x_1)\).
02

Understanding the undefined slope

From the equation in step 1, it can be observed that the denominator is \(x_2 - x_1\), which represents the horizontal change. When this is zero, it means there's no horizontal change, i.e., the x-values of both points being considered are the same. But mathematically, division by zero is undefined. Thus, when the slope is undefined, it implies that the line described is a vertical line.
03

Practical implications of an undefined slope

In graphs and charts, a vertical line represents a sudden and drastic change. For example, in a time-distance graph, a vertical line could represent teleportation (instantaneous change of position), assuming that time is represented on the X-axis and distance on the Y-axis. Note that this is a theoretical or hypothetical interpretation. Realistically, most natural phenomena are represented by defined slopes.
04

Summary

In conclusion, when the slope of a line is undefined, it indicates a vertical line, where there's no 'run' (horizontal change), and 'rise' (vertical change) can be any real number.

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Most popular questions from this chapter

$$\text { Solve: } 3 x+5=4(2 x-3)+7$$

What is an \(x\)-intercept of a graph?

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=-\frac{5}{2} x+1$$

A new car worth \(\$ 24,000\) is depreciating in value by \(\$ 3000\) per year. The mathematical model $$y=-3000 x+24,000$$ describes the car's value, \(y,\) in dollars, after \(x\) years. a. Find the \(x\)-intercept. Describe what this means in terms of the car's value. b. Find the \(y\)-intercept. Describe what this means in terms of the car's value. c. Use the intercepts to graph the linear equation. Because \(x\) and \(y\) must be nonnegative (why?), limit your graph to quadrant I and its boundaries. d. Use your graph to estimate the car's value after five years.

Solve for \(y\) and put the equation in slope-intercept form. $$y+3=-\frac{3}{2}(x-4)$$
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