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Chapter 11: Problem 231

In the following exercises, find the slope of each line. $$ y=3 $$

Short Answer

Expert verified
The slope of the line is 0.

Step by step solution

01

Understand the Line Equation

The given equation is in the form of a horizontal line: \(y = 3\).
02

Identify Characteristics of a Horizontal Line

A horizontal line has the same y-value for all x-values. In this case, it is always 3.
03

Determine the Slope of a Horizontal Line

The slope of any horizontal line is 0. This is because there is no change in the y-value as the x-value changes.

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Key Concepts

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

horizontal line
A horizontal line is a unique type of line in coordinate geometry. It is identified by its constant y-value regardless of the x-value. For example, in the equation \(y = 3\), the value of y is consistently 3 across all values of x.
This characteristic creates a line that runs parallel to the x-axis. In simple terms, every point on the horizontal line shares the same y-coordinate as specified in the equation. Because of this, horizontal lines are quite straightforward to identify and work with in various math problems.
Understanding this concept can simplify many tasks in coordinate geometry.
line equation
The line equation is a fundamental concept in algebra and geometry, describing a line on a coordinate plane. There are different forms of line equations:
  • Standard Form: \(Ax + By = C\)
  • Slope-Intercept Form: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept
  • Horizontal Line: \(y = c\), where \(c\) is a constant
  • Vertical Line: \(x = a\), where \(a\) is a constant
In our example of a horizontal line, the equation is \(y = 3\). This indicates that no matter what x-value we choose, y will always be 3. This type of line simplifies the graphing process because it only requires plotting a single y-value.
slope
The slope of a line is a measure of its steepness and direction. It is calculated as the ratio of the change in y (rise) to the change in x (run). Mathematically, it's expressed as:
\( \text{slope} = \frac{\Delta y}{\Delta x} \)
For a horizontal line like \(y = 3\), the y-value remains constant while the x-value changes. This results in:
  • \(\Delta y = 0\)
  • \(\Delta x\) is any non-zero value
Plugging these into the slope formula gives: \[ \text{slope} = \frac{0}{\Delta x} = 0 \] Thus, the slope of a horizontal line is always 0. This signifies that there is no vertical change as we move along the line, which fits the definition of a horizontal line perfectly.

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