Gradient of a straight line

The gradient of a straight line means the slope of the straight line. The gradient is the measure of the angle of a point of a straight line. A gradient can go upward, which means from left to right, or downward, which means from right to left. The gradient can be positive as well as negative, and it does not need to be a whole number.

E3.2A: Find the gradient of a straight line.

The gradient is the measure of the factor of a straight line. When the gradient of $$y$$ is small, the slope of the line will tend horizontal. If the gradient is large, the line will tend vertical. As the positive gradient value of $$y$$ increases, and the negative gradient value of $$y$$ decreases, $$x$$ will increase in both cases. 

A straight line is a set of points with a constant gradient between any two points. The formula for calculating the gradient of the straight line $$m$$ is $$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$. 

Worked example

 Example 1:  A horse gallops for $$15 \text{minutes}$$ and covers a distance of $$10 \text{kms}$$. First, convert this into an equation of a straight line and then locate the gradient of the line and describe what it shows.

Step 1: Plot the points on the graph and draw a straight line accordingly.

Step 2: Substitute the value of $$x_{1}=0$$, $$x_{2}=15$$, $$y_{1}=0$$ and $$y_{2}=10$$ in gradient formula.

The formula of the gradient is $$m=\frac{y_{2}-y_{1}}{ x_{2}-x_{1}}$$.

So, $$m=\frac{10-0}{15-0}$$.

Step 3: Solve the equation $$m=\frac{10-0}{15-0}$$.

$$m=\frac{2}{3}$$.

So, the gradient of the straight line is $$\frac{2}{3} \text{km} \setminus \text{min}$$. In this, it has been noticed that the distance-time graph provides the help of an equation that will represent the speed and distance covered by the horse as $$d=\frac{2}{3}t$$. This means that for every one minute the horse is galloping, it covers one and a half kilometers.

E3.2B: Calculate the gradient of a straight line from the coordinates of two points on it.

The gradient of a straight line is the rate at which the line is rising or falling vertically for every unit to the right (on the x-axis). If there are two points or coordinates given suppose $$(x_{1},y_{1})$$ and $$(x_{2},y_{2})$$, then the gradient of the line $$m$$ can be calculated as $$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$. 

In this, if the slope value is negative. It shows that the slope is going in the downwards direction, but if the value of the slope is positive, then it shows that the slope is going in an upward direction.

Worked example

Example 1: Find the gradient of the line passing through the points $$A(2,3)$$ and $$B(-3,0)$$.

Step 1: Plot the points on the graph and join them with a straight line.

Step 2: Substitute the value of $$x_{1}=2$$, $$x_{2}=-3$$, $$y_{1}=3$$ and $$y_{2}=0$$ in the gradient formula.

Formula of gradient is $$m=\frac{y_{2}-y_{1}}{ x_{2}-x_{1}}$$.

So, $$m=\frac{0-3}{-3-2}$$.

Step 3: Solving the equation $$m=\frac{0-3}{-3-2}$$.

$$m=\frac{3}{5}$$.

Therefore, the gradient of the straight line is $$\frac{3}{5}$$.