Geometry of the Line (2) The Equation of the Line

The equation of the (straight) line isn't difficult. In fact it's dead easy.

The reason is this. Any curve or line has a statement that describes what happens to a y-co-ordinate when you plug in a given x-value. This is why we talk of f(x), because the y co-ordinate is basically the function of the x under certain conditions, which differ for various lines and curves. With this statement you have the potential to describe the co-ordinates of the curve or line completely knowing only the x-value.

The equation is simply the statement of what happens to the y co-ordinate when you select any x co-ordinate you like. It contains the gradient of the line within it, because clearly where the y-value is will depend on how steep the line is.

We often hear about y=mx + c , where m=gradient, and c= y-intercept (or where the line crosses the y-axis).

It is fine and dandy, to be sure.

But for MPC1 purposes the following equation is also useful, not least because solving it leads to y=mx+c.

(y - y₁) = m(x - x₁)

Where (x,y) is any, unknown point on the line and (x₁, y₁) is a known point.



For example: take a line passing through (2,4) and (6,10).

Firstly, work out m. 6/4. or 3/2.

Plug that into the equation: (y-4) = 3/2(x-6) or y-4 = 3/2x - 9 or y= 3/2x -5.

Suddenly you have your y-intercept (-5) and your y =mx+c.

The only thing is, you need the gradient, so you need to know two points on the line from the start.