In the previous section, we have seen that the graph of a linear function f(x) = ax+b is a straight line which is inclined with certain sort of steepness. A measure of this steepness is the property called gradient or slope. In this section, we will discuss a way of finding the gradient of a line.

Note: In the sequel, we may use the term ‘slope’ for gradient, a’ line’ for straight line, and the symbol ∆ (read ‘delta’) means “the change in”. For instance, ‘∆x’ means ‘the change in x’.

From geometry, you know that two distinct points in a plane determine a unique line. So, any two distinct points (x₁, y₁) and (x₂, y₂) that lies on a line can be used to describe the line and to define the slope of the line.

For any two distinct points (x₁, y₁) and (x₂, y₂) on a line, we define

∆x = x₂ - x₁ (change in the values (coordinates) of x)

∆y = y₂ - y₁ (change in the values (coordinates) of y)

The slope, denoted by m, of a line L that passes through point (x₁, y₁) and (x₂, y₂) is defined by