Slope Formula
The slope of a line is a measure of its steepness or incline, calculated as the ratio of the vertical change to the horizontal change between two points on the line. This slope formula is commonly used in algebra and geometry to understand the direction and steepness of lines on a coordinate plane.
Formula for the Slope between Two Points
If we have two points \( (x_1, y_1) \) and \( (x_2, y_2) \) on a line, the slope \( m \) of the line passing through these points is given by the formula:
\( m = \dfrac{y_2 – y_1}{x_2 – x_1} \)
In this formula:
- \( y_2 – y_1 \) is the change in the \( y \)-coordinates, also called the “rise”
- \( x_2 – x_1 \) is the change in the \( x \)-coordinates, also called the “run”
- \( m \) represents the slope, indicating the direction and steepness of the line
A positive slope means the line rises as it moves from left to right, a negative slope means it falls, a slope of zero represents a horizontal line, and an undefined slope (where \( x_1 = x_2 \)) represents a vertical line.
Detailed Explanation of the Formula
The slope formula \( m = \dfrac{y_2 – y_1}{x_2 – x_1} \) calculates the ratio of the change in \( y \) to the change in \( x \), describing how much the line rises or falls for each unit it moves horizontally. This “rise over run” approach provides a measure of the line’s steepness.
- Calculate the difference in the \( y \)-coordinates: \( y_2 – y_1 \).
- Calculate the difference in the \( x \)-coordinates: \( x_2 – x_1 \).
- Divide the change in \( y \) by the change in \( x \) to find the slope \( m \).
Example 1: Calculating the Slope between Two Points
Problem: Find the slope of the line passing through the points \( (2, 3) \) and \( (6, 11) \) on a coordinate plane.
Solution:
- Write down the formula: \( m = \dfrac{y_2 – y_1}{x_2 – x_1} \).
- Substitute \( x_1 = 2 \), \( y_1 = 3 \), \( x_2 = 6 \), and \( y_2 = 11 \): \( m = \dfrac{11 – 3}{6 – 2} \).
- Calculate the differences: \( m = \dfrac{8}{4} \).
- Simplify: \( m = 2 \).
The slope of the line passing through the points \( (2, 3) \) and \( (6, 11) \) is \( 2 \), indicating a positive slope where the line rises by 2 units for each 1 unit it moves to the right.
Example 2: Slope of a Horizontal and Vertical Line
Problem 1: Find the slope of the line passing through the points \( (4, 5) \) and \( (8, 5) \).
Solution:
- Use the formula: \( m = \dfrac{y_2 – y_1}{x_2 – x_1} = \dfrac{5 – 5}{8 – 4} = \dfrac{0}{4} = 0 \).
A slope of \( 0 \) indicates a horizontal line, meaning no change in \( y \) as \( x \) increases.
Problem 2: Find the slope of the line passing through the points \( (3, 2) \) and \( (3, 10) \).
Solution:
- Use the formula: \( m = \dfrac{y_2 – y_1}{x_2 – x_1} = \dfrac{10 – 2}{3 – 3} = \dfrac{8}{0} \), which is undefined.
An undefined slope indicates a vertical line, meaning no change in \( x \) as \( y \) increases.
The slope formula is an essential tool in algebra and geometry, allowing us to calculate the steepness and direction of lines on a coordinate plane.