2 The gradient of a straight line through two points

A common requirement is to find the gradient of a line when we know the coordinates of two points on it. Suppose the two points are $A\left(x_1,y_1\right)$ , $B\left(x_2,y_2\right)$ as shown in the following figure.

Figure 25

The gradient of the line joining $A$ and $B$ can be calculated from the following formula.

Example 13

Find the gradient of the line joining the points $A\left(0,3\right)$ and $B\left(4,5\right)$ .

Solution

We calculate the gradient as follows:

Thus the gradient of the line is $\frac{1}{2}$ . Graphically, this means that when $x$ increases by $1$ , the value of $y$ increases by $\frac{1}{2}$ .

Task!

Find the gradient of the line joining the points $A\left(-1,4\right)$ and $B\left(2,1\right)$ .

Answer

Thus the gradient of the line is $-1$ . Graphically, this means that when $x$ increases by $1$ , the value of $y$ decreases by $1$ .

Exercises

1. Calculate the gradient of the line joining $\left(1,0\right)$ and $\left(15,-3\right)$ .
2. Calculate the gradient of the line joining $\left(10,-3\right)$ and $\left(15,-3\right)$ .

Answer