Problem 78 - Intermediate Algebra

Question

Find the slope of each line. The line with equation \(y=3\)

Step-by-Step Solution

Answer

The slope is 0.

Step 1: Identify the Type of Line

The equation given is in the form \( y = 3 \), which is a horizontal line. A horizontal line runs parallel to the x-axis.

Step 2: Understand Slope for Horizontal Lines

The slope of a line measures the steepness or incline of the line. For horizontal lines, there is no incline as the line does not rise or fall as you move left to right.

Step 3: Determine the Slope

For horizontal lines, the slope is always 0 because there is no vertical change as you move along the line. Therefore, the slope of the line with equation \( y = 3 \) is 0.

Key Concepts

Horizontal Line

A horizontal line is a straight line that travels from left to right without any vertical movement. In mathematical terms, a horizontal line is represented by an equation of the form \( y = c \), where \( c \) is a constant. Here, each point on the line has the same y-coordinate. This characteristic is significant because it aids in effortlessly identifying the line on a graph. For example, the line with the equation \( y = 3 \) is horizontal and intersects the y-axis at the point (0, 3). Additionally, a horizontal line features the qualities of:

• Being parallel to the x-axis
• Having the same y-coordinate for all points on the line
• Providing zero slope, which signifies no vertical change

Recognizing when you're dealing with a horizontal line helps in predicting its graphical behavior and calculating its properties.

Equation of a Line

The equation of a line is a mathematical statement expressing the relationship between the x and y coordinates on a graph. The most common form is the slope-intercept form, which is written as \( y = mx + b \). In this equation:

• \( m \) stands for the slope of the line, indicating its steepness and direction.
• \( b \) represents the y-intercept, or the point where the line crosses the y-axis.

For horizontal lines such as \( y = 3 \), the equation simplifies since there is no variable x; it resembles the equation \( y = c \), where \( c \) is a constant. This form emphasizes that every point on the line has the same y-coordinate. By understanding the components of a line's equation, you can swiftly identify its slope and intercepts, simplifying graph interpretations and analyses.

Understanding Slope

The slope of a line is a crucial concept in geometry and algebra, measuring how steep a line is. It is calculated as the 'rise over run,' which means dividing the vertical change by the horizontal change between two points on the line. In formula terms:\( m = \frac{y_2 - y_1}{x_2 - x_1} \)Where:

• \( y_2 \) and \( y_1 \) are the y-coordinates of two distinct points on the line.
• \( x_2 \) and \( x_1 \) are the corresponding x-coordinates.

For a horizontal line, such as one described by \( y = 3 \), there is no vertical change between any two points because the y-value remains constant. As a result, the slope is zero. This means that as you move along the line horizontally, there is no upward or downward motion, indicating the line is perfectly flat. Understanding how to calculate and interpret slope helps in analyzing the direction and inclination of various types of lines on a graph.