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Chapter 2: Problem 11

True or false: The slope between any two distinct points on a nonvertical line is a constant.

Short Answer

Expert verified
True. The slope of a nonvertical line is constant.

Step by step solution

01

Understand the Concept of Slope

The slope of a line measures its steepness and is calculated as the change in the y-values divided by the change in the x-values between two points (rise over run). If a line is not vertical, it has a defined slope.
02

Consider the Definition of a Nonvertical Line

A nonvertical line is a line that is not perpendicular to the x-axis. Hence, it has a run (change in x-values) that is not zero.
03

Calculate the Slope between Two Points on a Nonvertical Line

Given two points \( (x_1, y_1) \) and \( (x_2, y_2) \) on the line, the slope is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Since a nonvertical line has a constant rate of change in the y-values with respect to the x-values, this ratio remains unchanged regardless of the specific points chosen on the line.
04

Validate the Statement

As the slope remains constant for any two distinct points on a nonvertical line, the statement 'The slope between any two distinct points on a nonvertical line is a constant' is true.

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Key Concepts

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

constant slope
The concept of a constant slope is fundamental in understanding linear functions. The slope of a line, often denoted as \( m \), is a measure of how steep a line is. For a line to have a constant slope, it means that the steepness does not change no matter which two points on the line you choose to look at.
The slope is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Where:
  • \( y_2 \) and \( y_1 \) are the y-coordinates of two points on the line
  • \( x_2 \) and \( x_1 \) are the x-coordinates of those same two points

In simpler terms, the slope is the
nonvertical line
When we talk about a nonvertical line, we refer to any line that is not perfectly up and down. In other words, a nonvertical line moves both horizontally and vertically.
Because a nonvertical line has a horizontal movement (change in the x-values), the change in the x-coordinates \( x_2 - x_1 \) is never zero.
This ensures the denominator in the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) is never zero, making the slope a well-defined number.

Examples of nonvertical lines include lines that are flat (horizontal), diagonal, or any line that has a tilt, but not perfectly straight up.
rise over run
The phrase 'rise over run' is a simple way to remember how the slope of a line is calculated.

  • 'Rise' refers to the vertical change between two points on a line (the difference in the y-values: \( y_2 - y_1 \)).

  • 'Run' refers to the horizontal change (the difference in the x-values: \( x_2 - x_1 \)).

To visualize it:
Imagine walking up a gentle hill. The rise is how much you go up vertically; the run is how far you walk horizontally. When you divide the rise by the run, you get the slope, capturing both the direction and steepness of the line.

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