16. Slope of a Line

Hi. This lesson will talk about how to find the slope of a line.

We’ll consider two cases:

(i) Slope of a line whose equation is known

(ii) Slope of a line joining two points

Case 1

Let the equation of the line be ax + by + c = 0. Then, how do we find its slope?

We know that if the equation of the line looks like y = mx + c (slope-intercept form), then m is the slope of the line.

So, if we transform the first equation so that it looks like the second, then the coefficient of x will give us the slope.

Let’s do this!

The equation ax + by + c = 0 can be transformed as:

ax + by + c = 0

⇒ by = -ax – c

⇒ y = (-a/b)x + (-c/b)

This gives the slope of the line as -a/b (i.e., the coefficient of x). And we’re done!

We can remember this as a formula.

Slope = -(coefficient of x/coefficient of y)

For example, the slope of the line 2x – y + 10 = 0 will be -(2)/(-1) or 2.

This formula is valid only when the line is in the form ax + by + c = 0. If the equation is not in this form, say ax + by = c or ax = by + c, then we’ll bring all the terms to the LHS and then apply the above formula.

For example, to find the slope of the line 4x = y + 5, the equation should first written as 4x – y – 5 = 0. Then, the slope would be -(4)/(-1), or 4.

That’s all. Let’s move to the second case.

Case 2

I’ve already discussed this earlier. But it makes sense to discuss it here once again.

Let A(x₁,y₁) and B(x₂,y₂) be two points. We need to calculate the slope of the line joining A and B, or the slope of AB.

Recall that the slope of a line is the tangent of the angle made by the line with the X-axis (measured anticlockwise).

To find that, let’s construct AC parallel to the Y-axis and BC parallel to the X-axis.

Now, in triangle ABC,

tanθ = BC/AC

⇒ tanθ = (y₂ – y₁)/(x₂ – x₁)

⇒ m = (y₂ – y₁)/(x₂ – x₁)

And, we’re done!

A small point to note. If x₁ = x₂ (and y₁ ≠ y₂) then the denominator of the previous expression becomes zero, and the slope is undefined.

In this case, the line will be vertical, and the angle it makes with the X-axis would be 90°. And, tan90° is (also) undefined. It’s all connected!

Lesson Summary

1. The slope of the line ax + by + c = 0, is equal to -a/b.
2. The slope of the line joining the points (x₁, y₁) and (x₂, y₂) is equal to (y₂ – y₁)/(x₂ – x₁).

Remember these two, and you’ll be much comfortable later. In the next lesson, we’ll derive an expression to find the angle between two lines, whose slopes are known.