Question

The equation of a line is `y=mx+1`. How do you find the value of the gradient m given that `P(3,7)` lies on the line?

Answer 1

`m = 2`

Explanation:

The problem tells you that the equation of a given line in slope-intercept form is

`y = m * x + 1`

The first thing to notice here is that you can find a second point that lies on this line by making `x=0`, i.e. by looking at the value of the `y`-intercept.

As you know, the value of `y` that you get for `x=0` corresponds to the `y`-intercept. In this case, the `y`-intercept is equal to `1`, since

`y = m * 0 + 1`

`y = 1`

This means that the point `(0,1)` lies on the given line. Now, the slope of the line, `m`, can be calculated by looking at the ratio between the change in `y`, `Deltay`, and the change in `x`, `Deltax`

`m = (Deltay)/(Deltax)`

Using `(0,1)` and `(3,7)` as the two points, you get that `x` goes from `0` to `3` and `y` goes from `1` to `7`, which means that you have

`{(Deltay = 7 - 1 = 6), (Deltax = 3 - 0 = 3) :}`

This means that the slope of the line is equal to

`m = 6/3 = 2`

The equation of the line in slope-intercept form will be

`y = 2 * x + 1`

graph{2x + 1 [-1.073, 4.402, -0.985, 1.753]}