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Chapter 3: Problem 34

Find the equation, given the slope and a point. $$ m \text { undefined; }(3,0) $$

Short Answer

Expert verified
The equation is \( x = 3 \).

Step by step solution

01

Understanding the Given Information

We are given a slope that is undefined and a point on the line, (3,0). An undefined slope indicates a vertical line.
02

Determine the Equation of a Vertical Line

A vertical line passes through all points with the same x-coordinate. Since the point given is (3,0), the equation of the line will be x constant.
03

Write Down the Equation

For a vertical line passing through (3,0), the equation is simply \( x = 3 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope
The concept of slope is fundamental when talking about lines in geometry. Slope can be thought of as the measure of steepness of a line. In mathematical terms, the slope of a line is the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. This is often represented with the letter "m."
  • If a line rises from left to right, it has a positive slope.
  • If a line falls from left to right, it has a negative slope.
  • A horizontal line has a zero slope.
An undefined slope is unique as it only occurs in vertical lines, where changes in the y-values can be infinitely large while the x-values do not change. Thus, the "run" in the slope formula is zero, making its division undefined.
Vertical Line
Vertical lines are a fascinating aspect of geometry and algebra. These lines are characterized by having an undefined slope and always running parallel to the y-axis. The value of "x" remains constant while "y" can vary.

The equation of a vertical line is always in the form of \( x = a \), where "a" is the constant x-value for every point on the line. In our example, since the line passes through the point (3,0), the equation becomes \( x = 3 \). Here, it doesn't matter what the y-value is; as long as the x-value is 3, the point lies on the line.
  • Vertical lines do not intersect the y-axis at any single point, unlike non-vertical lines.
  • They represent infinite values moving upwards or downwards while having a fixed x-value.
Undefined Slope
An undefined slope arises when dealing with vertical lines. In the standard slope formula \( m = \frac{rise}{run} \), the run (or the change in x) is zero for vertical lines. Since division by zero is not possible in mathematics, the slope is termed undefined.

This characteristic helps to distinguish vertical lines from other lines, as it becomes impossible to express their steepness. Unlike horizontal lines, where the slope is precisely zero, vertical lines do not have a defined numerical value for steepness.
  • An undefined slope is key to recognizing the equation of a vertical line, \( x = a \).
  • It represents scenarios where the line extends infinitely along the y-axis with no horizontal variation.

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Most popular questions from this chapter

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