Pair of linear equation in two variables

Question

This is from a text book:-
"The general form of a linear equation in two variables is $ax + by + c = 0$ or, $ax + by = c$ where $a, b, c$ are real numbers such that $a ≠ 0$, $b≠0$ and $x, y$ are variables.
(we often denote the condition $a$ and $b$ are not both zero by $a^2+b^2≠0$.)"
I don’t understand this last condition.
How can we say that $a^2+b^2≠0$ represents the condition that $a$ and $b$ are not both zero.
Let $a = 0, b = 1$, then also this condition fulfills.
Any help?

Answer 1

$a$ and $b$ are both zero $\iff$ $a=b=0$, so
$a$ and $b$ are NOT both zero $\iff$ at least one of $a,b$ is not $0$
which is equivalent to $a^2+b^2=0$ in the case that $a,b$ are both real numbers.

Answering to your comment, yes, you are right. Maybe a better way to write in an inequality is $(a,b)\neq(0,0)$ instead of "$a\neq0,b\neq0$"

Answer 2

Generally words "together", "both" and comma sign "," is used for logical operation AND (conjunction), denoted by $x \land y$. So $$(a \ne 0, b \ne 0) \Leftrightarrow (a \ne 0 \land b \ne 0) \Leftrightarrow (a^2+b^2 \ne 0)$$