What is a linear equation?

Question

How do we define the linear equation? I mean, it looks like a polynomials with degree one but I'm not sure if $ax+by+c=0$ is a linear equation if $a=b=0$?

Answer 1

A linear equation is any equation that can be written in the form $$a_1x_1 + a_2x_2 + \cdots + a_nx_n = b$$ where $a_1, a_2, \ldots, a_n$, and $b$ are constants and $x_1, x_2, \ldots, x_n$ are variables without using multiplication or division to get it there.

So $ax + by + c = 0$ can be turned into $ax + by = -c$ and hence is linear. But $ax^2 + bxy + cx = 0$ needs division in order to turn it into $ax + by = -c$ (and that division changes the solutions) so it is not linear. The end result, as you point out, is that a linear equation is any equation given by a polynomial of degree at most 1.

The number of variables, $n$, is allowed to be zero if you like, as are all the constants $a_i$ and the constant $b$ ($0 = 0$ is linear). So in sum: yes, $ax + by + c = 0$ is linear even if $a = b = 0$.