Where are linear equations with large number of variables used? Do weather prediction / financial models or missile / rocket trajectory prediction model use these equations?
 What method or algorithm is used for the same?
 A: Where would linear equations not be used? For example in factorization of large integers? Well, there is Fermat's method: If yho have $x,y$ with $x^2\equiv y^2\pmod N$, then $N\mid (x+y)(x-y)$ so $\gcd(N,x\pm y)$ may happen to be proper divisors of $N$. So far, so good, but subtraczing (multiples of) $N$ from some  $x^2$ will hardly ever produce a square. But if you have found enough near-squares, i.e. $x_i^2-k_iN=y_i^2z_i$ where the $z_i$ contain only "some" small primes, then you may hope to combine these in a way to produce a perfect square. That is: We try to find a nontrivial element in the kernel of a matrix over $\mathbb F_2$ of size number-of-primes-called-small.
A: The first computer program to describe a linear model of the economy was written at Harvard by Wassily Leontief. It was written with 500 equations in 500 unknowns. That was too many for the computer so he broke it down into 42 equations with 42 unknowns, I don't know what algorithm he used to solve this but its probably worth looking it up.
A: Nonlinear systems of equations are solved using Newton's method. Guess what - at its core linear equations have to be sold. 
Any computation to solve problems with huge number of unknowns (weather prediction, earthquake inversions, any simulation of physical systems) has to solve linear equations. 
These equations are solved by iterative methods that do not require access to the complete matrix, but only need matrix times vector multiplications.
