I'm not really a maths guy, apologies if this is either trivial or impossible
If I have 2 formulas, say (ax + cxy) / (ax + b + cxy + d) = r and (cxy + dy) / (ax + b + cxy + d) = s
and I need to find both x and y, how would I go about this, can it be done?
I'm trying to find the adjustments needed to make 2 sets of properties fit desired ratios The first property would put it in either a or c if true, b or d if not The second property would put it in either a or b if true, c or d if not
Maybe this would work as an analogy Say you had 2 teams of people and wanted to work out wage adjustments as ratios needed so that Team 1 earned 60% of the total wages and Team2 40%, and the split between men and women was equal.
In this case A and C are Team 1, and C and D are women (I'm not sure if I'm confusing things by splitting into 4 groups?)
x would then be the multiplier for Team 1s wages and y would be the multiplier to womens wages
Notice that each ratio has the common denominator $ax + b + cxy + d.$
Therefore, its quite easy to subtract s from r giving
$$(r - s) = \frac{(ax + cxy) - (cxy + dy)}{ax + b + cxy + d} = \frac{ax - dy}{ax + b + cxy + d}$$
Then you can multiply and expand out the terms as follows:
$$(r-s)ax + (r-s)b + (r-s)cxy + (r-s)d = ax - dy$$
And solving for y looks something like this,
$$((r - s)cx + d)y = (r - s + 1)ax - (r - s)b$$
$$y = \frac{(r - s + 1)ax - (r - s)b}{(r - s)cx + d}$$
The variable x can be chosen arbitrarily, and y is given by the formula above. I'm not quite sure what you meant by the properties, but you'd certainly need something else to find exact values for x and y. Hope this helps