Question about formulas for straight lines

[Johnny5]

A student comes to you, and asks you where did they get the formula

 

y=mx+b from?

 

How do you answer the student?

 

Thank you

[uncool]

Say that by the definition of a line, since the increase is proportional to the x move, then it must be a constant plus a number times x. I.e. give them the basics of mathematical induction

If it works for some x-value a, then it must work for any x-value a + c.

Proof:

y(a) = ma + b

y(a+c)-y(a) must be proportional to c.

y(a+c)-y(a) = m*c for all c.

y(a+c) = m*c + y(a) = m*c + m*a + b = m*(a+c) + b.

Since only one formula can work for a function, if there is an a for which this works, this must be the formula.

It works for x = 0, y = b, so it must work for any value

-Uncool-

[Dave]

I would start by explaining the trivial case of y = x - i.e. 1 unit along is equivalent to 1 unit up. Then I'd explain the notion of the gradient, and how this affects the value of m. I'd conclude by adding in the constant to show how it shifts the line up and down.