Question 251662
I think I understand your question. Let's first solve for x and y in terms of a, b, c, d, u, and v.
multiply the first equation by c and the second equation by -a:
{{{c(ax + by = u)}}}
{{{-a(cx + dy = v)}}}
This will eliminate the x variable and we can solve for y.
{{{cby - ady = cu - av}}}
So, y equals
(i) {{{ y = (cu - av) / (bc - ad)}}}
substituting that into the first equation for x, we get
{{{ax + b(cu - av)/(bc-ad) = u}}}
Solving for x, we get
(ii) {{{X = (u - b((cu - av)/(bc-ad)))/a}}}
Now we have solve for X and Y in terms of a, b, c, d, u, and v. 
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Case 1: let d = a.
From (i and ii) we get
{{{X = (u - b((cu - av)/(bc-a^2)))/a}}}
{{{ y = (cu - av) / (bc - a^2)}}}
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case 2: let d = b.
From (i and ii) we get
{{{X = (u - b((cu - av)/(bc-ab)))/a}}} - -> {{{X = (u - ((cu - av)/(c-a)))/a}}}
{{{ y = (cu - av) / (bc - ab)}}}
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case 3: d = c.
From (i and ii) we get
{{{X = (u - b((cu - av)/(bc-ac)))/a}}}
{{{ y = (cu - av) / (bc - ac)}}}
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case 4: d = d
From (i and ii) we get
{{{X = (u - b((cu - av)/(bc-ad)))/a}}}
{{{ y = (cu - av) / (bc - ad)}}}

I hope that helps and is what you were looking for.