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Chapter 2: Problem 11

For each equation, find the slope. If the slope is undefined, state this. $$ y-2=6 $$

Short Answer

Expert verified
The slope is 0.

Step by step solution

01

Understand the equation

The given equation is: \( y - 2 = 6 \) This needs to be converted to the slope-intercept form of a line, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

Simplify the equation

Add 2 to both sides to isolate \( y \): \( y - 2 + 2 = 6 + 2 \) This simplifies to: \( y = 8 \)
03

Identify the slope

The simplified equation \( y = 8 \) represents a horizontal line where every point on the line has the same y-coordinate (8) and any x-coordinate. For a horizontal line, the slope is 0. Therefore, \( m = 0 \).

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Key Concepts

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

slope-intercept form
The slope-intercept form of a line is one of the most useful ways to represent a linear equation. This form is written as:

\( y = mx + b \)

In this equation:
  • \( y \) is the dependent variable
  • \( x \) is the independent variable
  • \( m \) represents the slope of the line
  • \( b \) stands for the y-intercept, which is the point where the line crosses the y-axis

Converting a given equation into slope-intercept form helps to quickly identify the slope and y-intercept. For example, in the given exercise, the equation was simplified to \( y = 8 \). This matches the slope-intercept form, where \( m = 0 \) and \( b = 8 \).
horizontal line
In geometry, a horizontal line is a straight line that goes from left to right or from right to left. It remains parallel to the x-axis. The equation of a horizontal line can be expressed as \( y = k \) where \( k \) is a constant value.

For instance, the equation \( y = 8 \) from the given exercise is a horizontal line. In this equation, the value of y is always 8, regardless of the value of x.

Some key properties of horizontal lines are:
  • All points on a horizontal line have the same y-coordinate.
  • A horizontal line is perpendicular to a vertical line.
  • The slope of a horizontal line is always zero.

Understanding horizontal lines helps in identifying the simplicity of several geometric and algebraic concepts.
slope of a line
The slope of a line, often denoted by the letter \( m \), measures the steepness and direction of the line. It is calculated as the change in the y-coordinates divided by the change in the x-coordinates between two points on the line. The formula for the slope is:

\[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \]
Some important points to remember about slope are:
  • If the slope (m) is positive, the line rises from left to right.
  • If the slope (m) is negative, the line falls from left to right.
  • If the slope (m) is zero, the line is horizontal, as seen in the exercise where \( m = 0 \) for the line \( y = 8 \).
  • If the slope (m) is undefined, it means the line is vertical.

The concept of slope is essential for understanding and analyzing the behavior of linear functions in algebra and geometry. It not only tells us how steep a line is but also in which direction the line moves.

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