Why we need the same number of equations and variables?

Question

I once read that to solve a system of equations, I need $n$ equations for $n$ variables.

I think I read it in a work by Euler. However, I have never managed to find a demonstration of this. So, why is this true? What is the demonstration for any system of equations? I would appreciate a formal proof and another intuitive one, thanks in advance.

Answer 1

It's not true in general.

Let $x_i \in \mathbb{R}$ where $i \in \{ 1, \ldots, 100\}$.

$$\sum_{i=1}^{100}x_i^2 =0.$$

We have $100$ variables and only one equation and we can solve it uniquely. That is $\forall i \in \{ 1, \ldots, 100\}, x_i=0$.