Slope Formula

In coordinate geometry finding the slope of a straight line is very important computation. In geometrical problem solving it is essential. The slope formula is used to calculate the steepness of the incline of a line with a curve. The x and y coordinate values of the lines are used for calculating the slope of the lines.  In this topic, the student will learn about slope and slope formula. Let us start learning!

The Slope Formula

What is the Slope?

The slope is computed as the ratio of the change in the y-axis to the change in the x-axis. The slope of a straight line will describe the line’s angle of steepness from the horizontal whether it rises or falls. When the line neither rises nor falls, then its slope will be zero. This is the case with a horizontal line where it extends infinitely to its left or right but without any sign of rise or fall.

In coordinate geometry, points are represented by a pair of coordinate values. This pair is having y to values x-value and y-value. For example (x,y) will represent a point on the geometrical plane. Also, (0,0) will represent the origin point. Let us have two points:

\( (x_1,y_1) and (x_2,y_2)  i.e. x_1, x_2 \) are the coordinates of x-axis and \(y_1, y_2\) are the coordinates of y-axis

On the geometrical plane.

The formula to calculate slope is defined as given below,

m= \( \frac{y_2-y_1}{x_2-x_1}\)

Where m is the slope of the line.

The numerical value for slope can be expressed as a ratio or fraction. The numerator will contain the difference of y-values, and the denominator will contain the difference of x-values. The above slope formula is conceptually defined as the rise overrun. This “rise” pertains to the movement of the point along the y-axis, and the “run” pertains to the movement along the x-axis.

Calculating the Slope

To find out the slope of a line we need only two points from that line, (x1, y1) and (x2, y2). There are three steps for calculating the slope of a straight line.

Step-1: Identify two points on the line.

Step-2: Select one to be \((x_1, y_1)\) and the other to be\( (x_2, y_2)\).

Step-3: Use the slope formula to calculate the slope.

Solved Examples

Example-1: Two points (15, 8) and (10, 7) are on a straight line. What is the slope of this line? Also show that points can be taken in any order.

Solution:

Step-1: Identify two points on the line.

In this example, we are given two points, (15, 8) and (10, 7), on a straight line.

Step-2: Select one to be \((x_1, y_1) and the other to be (x_2, y_2)\)

It does not matter which point we choose, so let’s take \((15, 8) to be (x_2, y_2). So, Let us take (10, 7) to be the point (x_1, y_1)\).

Step-3: Use the equation to calculate the slope.

After completing step 2, now we will calculate the slope using the equation for a slope:

m= \(\frac{8-7}{15-10} \)

i.e. m =\(\frac{1}{5} \)

Therefore slope is \(\frac{1}{5}\)

It really doesn’t matter which point we choose as first and which as second. We will show that this is true. Take this (15, 8) as \((x_2,y_2) and (10, 7) as (x_1,y_1)\)

Then substitute these values into the equation for slope:

m= \(\frac{7-8}{10-15}\)

i.e. m = \(\frac{-1}{-5} \)

i.e. m = \(\frac{1}{5}\)

Hence proved.