Estimate two variables given set of approximations

Question

I have a set of approximations of the form

known1 ~= 10X + 15YX - known2

where X and Y are unknown constants and known1 and known2 are known but different in every equation. For instance:

60 ~= 10X + 15YX - 40
50 ~= 10X + 15YX - 30
10 ~= 10X + 15YX - 20

(ignore the specific numbers, this is just to show which values change in each approximation)

I'm looking to get a good estimate of X and Y, ie I'm looking to minimize

sum of abs(10X + 15YX - known2 - known1) for all (known1, known2) pairs

Is there an analytical way to do this, or a more efficient way than trying random (X, Y) pairs?

Answer 1

Let your $z_i=known1_i + known2_i$ and $10x+15xy=p$.

You want to solve

$$\min_p \sum_{i=1}^n |p -z_i| $$

Of which we know that median minimize the quantity above.

Now that you have solved for $p$, you need to solve for $x$ and $y$.

$$10x+15xy=p$$

If you fix your $x$, you can solve for $y$. There is no unique solution.