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Chapter 3: Problem 81

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

Short Answer

Expert verified
The slope of the line y = 3 is 0.

Step by step solution

01

Identify the type of line

The equation given is in the form of y = k, where k is a constant. This represents a horizontal line.
02

Understand the properties of a horizontal line

A horizontal line has the same y-coordinate for all values of x. Since the y-coordinate does not change, the rise (change in y) is zero.
03

Calculate the slope

The slope of a line is given by the formula \( \text{slope} = \frac{\text{rise}}{\text{run}} \). For a horizontal line, the rise is zero and the run (change in x) can be any number. Therefore, the slope of a horizontal line is \( \frac{0}{\text{run}} = 0 \).

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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 line that runs left to right or right to left in the Cartesian plane. One key feature of a horizontal line is that its y-coordinate remains constant, no matter what the x-coordinate is. For example, in the equation \( y = 3 \), the y-coordinate for every point on the line is always 3, whether x is 1, -5, or 100.
Because the y-coordinate does not change, horizontal lines are quite simple to understand. They look like perfectly flat lines on a graph, without any upward or downward tilt.
slope calculation
The slope of a line measures its steepness. It's calculated using the formula: \[ \text{slope} = \frac{\text{rise}}{\text{run}} \]
The 'rise' represents the change in y-coordinates from one point to another, while the 'run' represents the change in x-coordinates. For a horizontal line like \( y = 3 \), the rise (change in y) is zero because the y-coordinate doesn't change at all. Since the rise is zero, the slope calculation for a horizontal line looks like this:
\[ \text{slope} = \frac{0}{\text{run}} = 0 \]
No matter what the run (change in x) is, dividing zero by any number still equals zero. Therefore, the slope of any horizontal line is always 0.
line equation
An equation of a line is a mathematical representation of all the points that lie on the line. For horizontal lines, the general form of the equation is \( y = k \), where \( k \) is a constant. This equation tells us that the y-coordinate is always \( k \), no matter what the x-coordinate is.
In our example, the equation is \( y = 3 \), meaning the y-coordinate is constantly 3 for all x-values. This is a quick way to identify the type of line and understand its slope without needing to plot it on a graph.

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