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Chapter 3: Problem 60

Find the slope of each line whose equation is given. If the slope is undefined, state this. $$ y=-4 $$

Short Answer

Expert verified
The slope is 0.

Step by step solution

01

Identify the Equation Type

The given equation is \( y = -4 \). This is a horizontal line because the equation is in the form \( y = c \), where \( c \) is a constant.
02

Understand the Slope of a Horizontal Line

For any horizontal line of the form \( y = c \), the slope is always 0. This is because there is no vertical change (rise) as you move along the line.
03

State the Slope

Since the equation is \( y = -4 \), which is a horizontal line, the slope is 0.

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Key Concepts

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

equation of a line
An equation of a line reveals important characteristics of that line, such as its slope and y-intercept.
In general, a linear equation can be written in the slope-intercept form as: \ [ y = mx + b\ ] where:
  • \(m\) is the slope of the line
  • \(b\) is the y-intercept, where the line crosses the y-axis

In the specific given exercise, we have the equation \ [ y = -4\ ]
This doesn't include an x-term. This tells us that no matter what value x takes, y will always be -4. This means the line is horizontal.
horizontal line
A horizontal line runs parallel to the x-axis and has a unique form of equation: \ [ y = c\ ]
Here, \(c\) is a constant. In this scenario, the value of \(c\) determines the vertical position of the line. Since the line does not rise or fall as x changes, we conclude that there is no vertical movement.

For the equation \ [ y = -4\ ]
  • The constant \( -4 \) means that the line crosses the y-axis at -4
  • No matter what x is, y always remains constant at -4
This is a horizontal line.
slope calculation
In mathematics, the slope of a line is a measure of its steepness and direction. We can find the slope by the ratio of the change in y to the change in x, often expressed as: \ [ m = \frac{{\Delta y}}{{\Delta x}}\ ]
For a horizontal line like \ [ y = -4\ ], there is no vertical movement; thus, \(\Delta y = 0\). Since there is no rise, the slope \( m \) becomes: \ [ m = \frac{{0}}{{\Delta x}} = 0\ ]
This simplifies to 0, meaning horizontal lines always have a slope of 0. This is because there is no rise or fall, just a constant y-value for all x-values.

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