Why do we add error in the definition of LWE? One of the various equivalent definitions of the LWE problem is the following:
Let $n,q$ be integers ($q$ usually is a prime number), $\chi$ a discrete probability distribution over $\mathbb{Z}$ (usually a discrete Gaussian distribution) and $s$ a secret vector from $\mathbb{Z}_q^n$.
We denote $\mathcal{L}_{s,\chi}$ the probability distribution over $\mathbb{Z}_q^n \times \mathbb{Z}_q$ obtained by choosing $a \in \mathbb{Z}_q^n$ uniformly at random, choosing $e$ uniformly at random from $\chi$ and considering it in $\mathbb{Z}_q$, and calculating $(a,b=(\langle a,s\rangle + e)) \in \mathbb{Z}_q^n \times \mathbb{Z}_q$.
The Search Learning With Errors is to recover $s$ from samples $(a,b)$ obtained from $\mathcal{L}_{s,\chi}$.
My questions is why do we add the error $e$? I suppose that it is for security reasons. In that case, how we would be able to obtain $s$ from $(a,b=(\langle a,s\rangle))$?
Since adding some error $e$ is a central block in a lot of lattice-based constructions, this could be a more general question. What is the point of adding this error?
In that case, how we would be able to obtain $s$ from $(a,b=(\langle a,s\rangle))$?

The operation $\langle a,s\rangle$ is a matrix multiplication, that is, completely linear, and hence Gaussian elimination allows us to recover $s$ efficiently.