Slope of a Line


Slope of a Line

Hi. This very small 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 (we could find the equation first and then use the first method, but that’s really not necessary)

Lets consider the first case.

Let the equation of the line be ax + by + c =0 (general form). 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 in the given equation into the slope-intercept form, then the coefficient of x will give us the slope.

Let’s do this !

The equation ax + by + c =0 can be transformed as y = (-a/b)x + (-c/b). This gives the slope of the line as -a/b. And we’re done!

Therefore the slope of a line (when expressed in the general form) will be -a/b or more generally  -(coefficient of x/coefficient of y)

Remember always to transform the equation in the general form, before using this expression.

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

And, the slope of the line 4x = y + 5 will not be (-4)/(1) ! Because it isn’t expressed in the general form.

The equation should first written in that form as 4x – y – 5 = 0, and now the correct slope would be -(4)/(-1), which is 4.

That’s all there to it.

Moving on to the next case.

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

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

Here is the figure which I used earlier..

Coordinate Geometry Slope of a line joining two points

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

The line joining A and B is extended to meet the X-axis, as shown. In triangle ACB, tanθ = BC / AC, which equals (y2-y1)/(x2-x1). (details of that here)

Well, tanθ is nothing but the slope. So, we’re done here. The slope of AB equals (y2-y1)/(x2-x1). (i.e. the ratio of the difference of the y coordinates to the difference of the x-coordinates)

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.

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