Mathematica: Solve function problem

Question

I have entered the following commands in Mathematica:

S = Sum[

  (y[i] - a *x[i] - b)^2,

  {i, 1, n}

  ]

Equation1 = D[S, a] == 0;

Equation2 = D[S, b] == 0;

Solve[Equation1 && Equation2, {a, b}]

The result of the last operation, the Solve, was nothing: {}

However all the steps before Solve operation were correctly calculated. If I put a certain value in the $n$ variable, so that the sum becomes with defined limit (for example $i=1..n, n = 10$) then everything is solved correctly.

What is the problem here and how to resolve it?

The program I made was simple Least Squares method. I just wanted to see the result in case when the variable $n$ is not fixed to some value.

EDIT1: The result of the Solve function if I set the $n = 2$ will be:

$\left\{\left\{a\to -\frac{-y[1]+y[2]}{x[1]-x[2]},b\to -\frac{x[2] y[1]-x[1] y[2]}{x[1]-x[2]}\right\}\right\}$

EDIT2: In case of not fixed $n$ value the result would be:

But I get empty list instead.

Here is what Equation1 and Equation2 contains if the $n$ value is not fixed:

$\sum _{i=1}^n -2 x[i] (-b-a x[i]+y[i])==0$

$\sum _{i=1}^n -2 (-b-a x[i]+y[i])==0$

So the Mathematica must handle this, but why it can't?