Slope Formula

Slope Formula is used to calculate a line’s inclination or steepness. It can be used to calculate the slope of any line by dividing the change in the y-axis by the change in the x-axis. The slope of a line is defined as the change in the line’s “y” coordinate with respect to the change in the line’s “x” coordinate. Finding the ratio of “vertical change” to “horizontal change” between any two distinct points on a line yields the slope. When the ratio is expressed as a quotient (“rise over run”), the same number is provided for every two distinct points on the same line. Learn more about slope formula in this article.

What is Slope?

Slope of a line determines its “steepness.” It is commonly represented by the letter m. As a result, the slope of a line is defined as the change in Y divided by the change in X. Because the change in Y is so large, the slope can be anything from zero to any number. However, the maximum slope is usually positive or negative infinity. Because the change in x is much smaller than the change in y, the change in x is much less than the change in y.

The slope of a line indicates how slanted the line is, comparing how much the line rises vertically to how much it runs horizontally.

Slope Formula

In coordinate geometry, the slope (often denoted as \( m \)) is calculated using the coordinates of two distinct points on the line. The formula for the slope of a line passing through points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[ m = \frac{y_2 – y_1}{x_2 – x_1} \]
Where,

\( (x_1, y_1) \) and \( (x_2, y_2) \): These are the coordinates of two distinct points on the line.
\( y_2 – y_1 \): This represents the vertical change (rise) between the two points.
\( x_2 – x_1 \): This represents the horizontal change (run) between the two points.

Derivation of Slope Formula

The slope of a line is a measure of its steepness and is commonly denoted by the letter m. The slope can be derived using two points on the line. Here’s a step-by-step derivation of the slope formula:

Step 1. Identify two points on the line:
Let the coordinates of the two points be \((x_1, y_1)\) and \((x_2, y_2)\).

Step 2. Understand the change in coordinates:
The change in the x-coordinates (horizontal change) is given by \( \Delta x = x_2 – x_1 \).
The change in the y-coordinates (vertical change) is given by \( \Delta y = y_2 – y_1 \).

Step 3. Define the slope:
The slope \( m \) of the line is defined as the ratio of the vertical change to the horizontal change between the two points. Mathematically, this is written as:
\[
m = \frac{\Delta y}{\Delta x}
\]

Step 4. Substitute the changes in coordinates:
Substitute the expressions for \( \Delta y \) and \( \Delta x \) into the slope formula:
\[
m = \frac{y_2 – y_1}{x_2 – x_1}
\]

The slope formula is:
\[
m = \frac{y_2 – y_1}{x_2 – x_1}
\]

Slope Equation

The slope is calculated as the ratio of the y-axis change to the x-axis change. The slope of a straight line describes the angle of steepness from the horizontal, regardless of whether the line rises or falls. When the line does not rise or fall, its slope is zero. This is the case with a horizontal line, which extends indefinitely to the left or right but shows no indication of rise or fall.

The slope equation, often referred to as the equation of a line in slope-intercept form, is a way to express a linear equation using the slope of the line and its y-intercept. The general form of this equation is:

y = mx + b

where:
y is the dependent variable (the value on the y-axis),
x is the independent variable (the value on the x-axis),
m is the slope of the line,
b is the y-intercept (the point where the line crosses the y-axis).

To derive the slope-intercept form, we start from the basic definition of the slope and use a known point on the line.

Step 1: Start with the slope formula:
\[
m = \frac{y_2 – y_1}{x_2 – x_1}
\]

Step 2: Let \((x_1, y_1)\) be a specific point on the line, and \((x, y)\) be any other point on the line:
\[
m = \frac{y – y_1}{x – x_1}
\]

Step 3: Solve for \( y \):
\[
y – y_1 = m(x – x_1)
\]
This is the point-slope form of the equation of a line.

Step 4: Isolate \( y \) to convert to slope-intercept form:
\[
y = m(x – x_1) + y_1
\]
Simplifying further, we get:
\[
y = mx – mx_1 + y_1
\]
\[
y = mx + (y_1 – mx_1)
\]

Step 5: Identify the y-intercept \( b \):
The term \( (y_1 – mx_1) \) represents the y-intercept \( b \). Therefore, we can rewrite the equation as:
\[
y = mx + b
\]

How to Calculate Slope?

To determine the slope of a line, we only need two points from that line, (x1, y1) and (x2, y2). The slope of a straight line is calculated in three steps.

Step 1. Determine two points on the line.

Step 2: Choose one to be (x1,y1) and another to be (x2,y2).

Step 3: Calculate the slope using the Slope Formula.

Examples Using Slope Formula

Example 1: Find the slope of the line passing through the points (1, 2) and (4, 6)

Solution :
Identify the coordinates: \( (x_1, y_1) = (1, 2) \) and \( (x_2, y_2) = (4, 6) \).
Calculate the difference in y-values: \( y_2 – y_1 = 6 – 2 = 4 \).
Calculate the difference in x-values: \( x_2 – x_1 = 4 – 1 = 3 \).
Divide the differences: \( m = \frac{4}{3} \).

So, the slope of the line is \( \frac{4}{3} \).

Example 2: Find the slope of the line passing through the points \( (-3, 5) \) and \( (2, -1) \).

Solution :
Identify the coordinates: \( (x_1, y_1) = (-3, 5) \) and \( (x_2, y_2) = (2, -1) \).
Calculate the difference in y-values: \( y_2 – y_1 = -1 – 5 = -6 \).
Calculate the difference in x-values: \( x_2 – x_1 = 2 – (-3) = 2 + 3 = 5 \).
Divide the differences: \( m = \frac{-6}{5} \).

So, the slope of the line is \( -\frac{6}{5} \).