When I search up the equations for Lagrange Multiplier, I usually get two different cases where one is $\nabla f(\mathbf{x}) + \nabla\lambda g(\mathbf{x}) = 0$ and the other one being $\nabla f(\mathbf{x}) - \nabla\lambda g(\mathbf{x}) = 0$. I have received answers that the sign change gets negated by the multiplier, but I don't really understand what that means.
How are both equations equivalent?
Conceptually, the $\lambda$ is just some proportionality constant to indicate that $\nabla f$ and $\nabla g$ are parallel. Two vectors $v$ and $w$ are parallel if $v=cw$ for some constant $c\in \mathbb{R}$. It does not have to be positive (and it shouldn't have to be, think about $v$ and $-v$!).