Maths Times Infinity > Topics > Linear

Linear Functions

Gradient

#y=mx+b#.

Where #m# is the gradient and #b# is the #y#-intercept.

#m# is the gradient because it's the amount #y# changes by when you change x. #b# is the y-intercept because when #x=0#, #y=b#. This is a very form because it has y as the subject. You can easily sub in values of #x# and find out what #y# is at that point.

Equation of a Line Given the Gradient and a Point

Given a gradient and a point, there is exactly one line which passes through it. This is intuitive - you start somewhere and you know which direction you're going in.

There's a formula for the equation of the line:

#y-y_1=m(x-x_1)#.

It can also be written like this:

#y=y_1+m(x-x_1)#.

You can think of it as y starts at #y_1# and then moves at a rate of #m# depending on the distance of #x# from #x_1#.

Another way of writing it:

#(y-y_1)/(x-x_1)=m#.

The gradient is m so for any point on the line #(x,y)#, the rise/run ratio from #(x_1,y_1)# to #(x,y)# should be #m#.

Equation of a Line Given Two Points

(y-y₁)/(x-x₁)=(y₂-y₁)/(x₂-x₁).

Same as Point and Gradient except first we calculate the gradient using the two points, then use either point as a kind of pivot.

Point of Intersection of Two Lines

Basically, you solve the two equations simultaneously.