estimate of solution for equation system with a "solutions pool"

hello everyone,

first, I have to say sorry, I am a programmer, not that much a mathematician,
so it may become difficult for me to articulate the problem that is bothering me currently.

here it is:
there is an equation system of, let's assume 1000, equations of the form:
FACTOR1 * Input1 + FACTOR2 * Input2 + ... FACTOR999 * Input999 = Solution

now, the deal is, there are no given certain values for the equations solutions.

instead, in the example, there is a pool of 10.000 values that come close
to be solutions for the equations, i.e. for each equation there are by average 10 approx. values.

how can I 'solve' the system for the FACTORs ?

what comes to my mind would be a backpropagation type algorithm,
but I like to know if there is something more efficient.

thanks