Working out equations of a line and calculating gradient

Straight line graphs.

All straight line graphs have an equation that can be written in the form 𝑦 equals 𝑚𝑥 add 𝑐, where 𝑚, the coefficient of the 𝑥 term, is the gradient or slope of the line. The constant, 𝑐, is the 𝑦-intercept. This is the value where the line crosses the 𝑦-axis.

Let's look at an example question.

A straight line passes through the points (1, 5) and (3, –1). Find the equation of this line.

First, draw a sketch to help make sure your answers are sensible. This doesn't need to be accurate, but you should mark the two given points in roughly the right positions, in the correct quadrants.

Remember, the first coordinate represents movement along the 𝑥-axis, left or right, and the second coordinate represents movement along the 𝑦-axis, up or down.

To finish the sketch, draw a straight line that goes through each of the points and the 𝑦-axis.

Then, you need to find two pieces of information, the gradient, 𝑚, and the 𝑦-intercept, 𝑐.

To find the gradient, you need to calculate the change in 𝑦 divided by the change in 𝑥.

It can be helpful to label the two points as A and B here.

The change in 𝑦 is given by the 𝑦 coordinate of point B, 𝑦B, subtract the 𝑦 coordinate of A, 𝑦A. Then, the change in 𝑥 is given by the 𝑥 coordinate of point B, 𝑥B subtract the 𝑥 coordinate of A, 𝑥A.

It doesn't matter which way round you label the points. You'll get the same answer if you chose to subtract the coordinates of point B from the coordinates of point A.

Labelling (1, 5) as point A and (3, –1) as point B, and using 𝑦B subtract 𝑦A over 𝑥B subtract 𝑥A, this gives a change in 𝑦 of –1 subtract 5, and a change in 𝑥 of 3 subtract 1.

Simplified, this is –6 divided by 2, which equals –3. So, the gradient, 𝑚, equals –3.

You can substitute this into the general equation for a straight line, 𝑦 equals –3𝑥 add 𝑐.

Now you've worked out the gradient, don't forget to check your answer makes sense.

The sketch shows that the 𝑦 value decreases as the 𝑥 value increases, meaning the line slopes downhill from left to right. This means the gradient is negative, which matches the calculated gradient of –3.

Next, to find the 𝑦-intercept, 𝑐, choose one of the given points, for example (1, 5), and then substitute 𝑥 equals 1 and 𝑦 equals 5 into the line’s equation, with the variable 𝑐 staying unknown for now. Doing this gives 5 equals –3 times 1 add 𝑐, which simplifies to 5 equals –3 add 𝑐.

Finally, solve this equation for 𝑐 so that you have all the information needed to find the equation of the line. So, add 3 to both sides to isolate 𝑐 on the right-hand side.

5 add 3 equals 8, so 𝑐 equals 8 and the equation of the line is 𝑦 equals –3𝑥 add 8.

To check, look at your sketch. The line crosses the 𝑦-axis at a positive 𝑦-value that's greater than 5, so 8 is a realistic answer for the 𝑦-intercept.

Finding the gradient.

Any straight line graph has a constant gradient, which is calculated by the change in 𝑦 divided by the change in 𝑥, along any section of the graph.

The gradient is measuring the steepness of the graph, so a constant gradient means the steepness stays the same across the entire graph.

Before calculating the gradient, consider if it will be positive or negative.

To do this, imagine walking along the line from left to right. You would be going uphill, which means the gradient is positive.

Then choose any two coordinates on the line, for example (2, 3) and (8, 6), and then work out the change in 𝑦 divided by the change in 𝑥.

So, the gradient is one half.

Now, what if we draw another line?

Again, consider if the gradient will be positive or negative. This time, if you are walking along the line from left to right, you would be walking downhill, so the gradient will be negative.

Then, choosing two random points on the line and using the same method as before, we can work out that the gradient is –2.