# Understand slope formula and Derivation? How to Find Slope with Two Points, Graphs and Equations

In this article, you will learn what is the slope of a line and its formula. You will also learn what are 3 slope formulas.

## What is Slope?

The slope is a geometric term that explains the inclination and steepness of a line. It tells how steep a line is and how quickly it increases or decreases. It is an useful idea in mathematics that finds the behaviour of straight lines. In a two-dimensional plane, the slope of a line is the ratio of differences along the x and y-axis.

It is a numerical measure of a line’s inclination and steepness with respect to horizontal and vertical axis. It is an important factor in geometry as well as in calculus.

## Slope Formula

As it is clear by the definition that the slope of line is the ratio of vertical and horizontal distance. So for a line y=mx+c, the general formula of slope is:

It helps to find an equation of a straight line. Here the slope is denoted by ‘m’.

Since the slope is the rate of change in vertical and horizontal coordinates, the formula can also be written as:

$$m \;=\; \frac{\Delta y}{\Delta x}$$

## Derivation of Slope Formula

Consider the equation of line,

$$y \;=\; ax \;+\; b$$

Suppose, (x1,y1) and (x2,y2) are two points on the line so,

$$y_1 \;=\; ax_1 \;+\; b$$

and

$$y_2 \;=\; ax_2 \;+\; b$$

By simplifying the above equations for a, we get,

$$a \;=\; \frac{y_2-y_1}{x_2-x_1}$$

Which is the equation of slope of the line y = ax + b.

## How to Find the Slope of a Line?

Suppose, a line is formed by joining two point A and B having coordinates (x1,y1) and (x2,y2) then the slope of line AB can be calculated as:

$$m \;=\; \frac{y_2-y_1}{x_2-x_1}$$

Here we provide you a step-by-step method to find the slope of a line. Use the following steps to find slope of a line:

• Find the change in vertical coordinates of the line.
• Find the change in horizontal coordinates.
• Substitute the values of change in vertical and horizontal coordinates in the slope formula, m = y2-y1/x2-x1
• Simplify the ratio to get the exact slope.

Let’s understand how to find the slope of a line by discussing slope formula examples.

## How to Find Slope on a Graph?

It is another method of finding slope of line from a graph. You can calculate the slope from a graph by using the following steps.

• Draw a line with random points on the graph.
• Label the point as A and B.
• Now calculate the rise from A to B vertically. Remembering the positive sign indicates upward direction and negative indicates downward direction.
• Now calculate the run from A to B which is the horizontal distance between them.
• Now use the divide the vertical distance (rise) by the horizontal distance (run) between them as, $$m \;=\; \frac{rise}{run}$$

## Related Formulas

In geometry, there are three major slope formulas. These are:

1. Slope-intercept Formula:

y = mx + c

2. Point slope Formula:

y-y1=m(x-x1)

3. Standard Form Formula:

Ax + By = C

## What is the Two Point Slope Formula?

Suppose, a line is formed by joining two point A and B having coordinates (x1,y1) and (x2,y2) then the two point slope of line AB can be calculated as:

$$m \;=\; \frac{y_2-y_1}{x_2-x_1}$$

## What is the Slope of a Line?

It is a numerical measure of a line’s inclination and steepness with respect to horizontal and vertical axis.

### Shaun Murphy

Last updated: July 07, 2023

I'm graduated in biomedical and electrical engineering. Specialization in sports and medical topics but will gladly tackle everything you throw at him. He is a sailor, hiker, and motorcyclist in his free time.