Slope Formula Explained With Examples

As per Mathematics, slope can be defined as a ratio of the change in altitude to a change in the horizontal distance between two points. The former is often referred as 'rise' and the later is known as 'run'. In short, this formula assesses steepness or an incline or a grade. A slope is often referred to as a gradient in Math.

A straight line is a one which joins two distinct points. For instance, take the example of the staircase of your home. It is the simplest and best example of Math in everyday life. It is a slope and if you view it geometrically or on graph, you can make out the axes clearly.

The line along the x axis of your graph is a horizontal line and the line which is aligned along the y axis, is a vertical line. So, work on a similar but small version of this image on the graph paper. Select two points on the axes and join them by a straight line. Now you have the x and the y coordinate values.

Measure the y coordinate axes which would be say, y₂ - y₁. Similarly, the x coordinate axes value would be x₂ - x₁.

So, the formula of slope between two points can be calculated as:

m = (y₂ - y₁) / (x₂ - x₁)

where m is the gradient of the image.

Just imagine, if you were required to calculate the slope of a huge landscape and you are not sure how, this method would make your problems a lot simpler. A point slope method helps in graphing a linear equation on a graph and when having graphed it, all you have to do is make pairs of the x and y coordinates and plot them.

The name itself implies that the slope is being calculated by using a single point on the graph. The given formula of slope between two points can be further modified as:

m = (y₂ - y₁) / (x₂ - x₁) = ∆y/∆x (rise/run),

where ∆ is the change indicated in the altitude and the horizontal distance.

So, the standard point slope equation is:
(y₂ - y₁) = m (x₂ - x₁)

If one goes by the Trigonometry/Calculus methods, it becomes all the more interesting to calculate the formula for slope. Here, the slope is associated with the angle formed by the slope. The angle is known as θ which can be expressed as:

m = tan θ = Sin θ/Cos θ,

A slope of 2 is given and the point for slope is (4,3) (x,y format). Calculate the equation in point slope form using the formula.

Solution:

We know, the slope m = (y₂ - y₁) / (x₂ - x₁) = ∆y/∆x
=> (y₂ - 3) = 2 (x₂ - 4)
=> y - 3 = 2 (x - 4)

A slope is formed by an angle of 30º. What is the gradient?

Solution:

m = tan θ = tan 30º = 0.5773 (using a scientific calculator)

This formula of slope is very important in calculation of tough arithmetic and algebraic equations in Astronomy and Physics, as it makes the job a lot easier. Understand this formula and apply it correctly and be assured, Math will be a fun subject for you!