Usually linear equations are in the form of

*y*= m*x*+ c (or can be rearranged to take this form).- m in this equation refers to the gradient, which is the steepness of the line; a positive value for m means that the line is upwards sloping and a negative value for m means that the line is downwards sloping.
- c in the equation is the
*y*intersect and it is the value at which the line crosses the*y*axis. c is the*y*intercept because the*x*coordinate for any point on the*y*axis is 0. This means that when we sub*x*as 0 into any line in the form*y*= m*x*+ c, we are only left with c because m is multiplied by zero, which means that it disappears.

**Example 1**

Suppose that we have a line with the equation y = 3x + 4, what is the

*y*intercept?

We can find the

*y*intercept by subbing in

*x*as 0 into the equation of the line.

Therefore, the coordinates for the y intersect is (0, 4). Alternatively, because the equation was in the form of

*y*= m*x*+ c, the*y*intercept would have been the value of c, which is 4, thus meaning the coordinates of the*y*intercept are (0, 4).**Changing the Value of c**

I am now going to have the line

*y*=

*x*and change the value for c. Currently, the value of c is zero and this means that the

*y*intercept has the coordinates (0, 0); the graph passes through the origin. Below is a table working out the coordinates of the line

*y*=

*x*.

I am now going to change the value of c in the graph to

*y*=*x*+ 1. The table and graph are shown below (the graph has the line*y*=*x*shown as well).From the graph, we can see that all of the points on the line

Let’s now plot the line

*y*=*x*have moved up by 1. The*y*intercept for the line*y*=*x*+ 1 is (0, 1).

Let’s now plot the line

*y*=*x*– 3. This line will intercept the*y*axis at -3 and it will have a gradient of 1. This means that we will move all of the coordinates from*y*=*x*down by 3.Finally, let’s plot the graph of

*y*=*x*+ 5. The*y*intercept for this line is 5 and the gradient of the line is 1. Therefore, all of the points on the graph of*y*=*x*move up by 5.**Example 2**

What is the

*y*intercept for the equation below?

We can find the

*y*intercept by subbing in*x*as 0.The

Alternatively, because the graph is already in the form of

*y*intercept is (0, -7).

Alternatively, because the graph is already in the form of

*y*= m*x*+ c, the*y*intercept is going to be the value of c. For the graph in this example, the value of c is -7, which means that the coordinates of the*y*intercept are (0, -7).**Example 3**

Find the gradient and

*y*intercept of the equation below. Also, roughly sketch what the graph looks like.

We are able to quickly find the gradient and

*y*intercept of a line by getting the line into the form*y*= m*x*+ c. Therefore, we need to rearrange the equation above and make*y*the subject. We do this by getting all of the*y*’s by their self on one side of the equation and all of the other terms on the other side of the equation. There are only*y*’s on the left and it makes sense to keep the*y*’s on this side. Therefore, we need to get the 12 from the left to the right. We move the 12 to the other side by doing the opposite; we take 12 from both sides.We want

*y*not 5*y*, so we divide both sides by 5.We now have the equation in the

Below is a quick sketch of what this graph looks like.

*y*= m*x*+ c form. The gradient of the line is -2 (the value for m), which means that the line is downwards sloping and fairly steep. The*y*value for the*y*intercept is 1 (the value for c), which means that the coordinates for the*y*intercept is (0, 1).Below is a quick sketch of what this graph looks like.