Gradient of a Straight Line Joining Two Points

Understanding the Concept

In Rectangular Cartesian Coordinates, the gradient of a straight line connecting two points can be calculated using a simple formula derived from the coordinate geometry.
Given two points, P(x1, y1) and Q(x2, y2), the gradient, often denoted as m, of the straight line joining P and Q is calculated by the formula:

m = (y2 - y1) / (x2 - x1)

The division operation inside the formula computes the ratio of the difference in y-coordinates to the difference in x-coordinates of the points P and Q.
The expression (y2-y1) represents the vertical change (along the y-axis), commonly referred to as “rise”, between P and Q; (x2-x1) stands for the horizontal change (along the x-axis), often termed as “run”.
The ratio of “rise” to “run”, i.e., rise/run, gives us the slope or gradient of the straight line connecting P and Q.

For example, if you wish to find the gradient of the line joining points A(3,2) and B(7,6), substitute these values into the formula: m = (6 - 2) / (7 - 3) = 4 / 4 = 1
If you’re given the gradient and one point, you can find the coordinates of another point on the line by rearranging the gradient formula and solving for the unknown coordinates.

The gradient of a straight line is a fundamental concept in coordinate geometry.
Truly understanding the principle behind the formula, rather than only committing it to memory, sets you up for success in manipulating it under varying situations.