Question

What similarities and differences are there between functions and linear equations? Are all linear equations functions? Is there an instance when a linear equation is not a function?

Answer 1

A function is a rule that explains how one variable can be computed from one or more other variables. A linear equation of the form y = mx + b is a function.

If you have an equation without ONLY one variable on one side, it is technically not a function, in my opinion.

b = mx - y and m = y/x are not explicit functions as written, if m and b are constants, but can be rewritten as linear functions. Those might be called implicit functions.

I think one get lost in semantics with questions like this. I wonder whether it really helps in teaching math.

Answer 2

With a function, for every X there is only one Y. With linear equations, for every X, there can be more than one Y.