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I'm in a little bit of a pickle here, can someone tell me if it is actually possible to solve these? From the general rule of thumb, 4 variables, 4 equations should be able to right?

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    Mosquito-nuking solution: the Jacobian determinant for this system of equations is zero, so no.2010-11-15
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    @Rainbow: offhand question, have you taken up calculus already? The "nuke" solution becomes less of a "nuke" if you have a lot of simultaneous equations.2010-11-15

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The best you can do is $20x = 21y$ and $21a = 20b.$ If you write your second pair of equations as $$a \left( \frac{20}{21} x \right) = 4000,$$ $$b \left( \frac{20}{21} x \right) = 4200$$ we note that they are equivalent to your first pair $$ax = 4200$$ $$bx = 4410,$$ simply by multipling up by the $20/21.$

As for your general rule of thumb, note that a system of equations can be inconsistent.

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    What if I can introduce more equations? Such as xc = 420, yc = 400? Or will it never satisfy?2010-11-15
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    @RainbowTheory Again, they give $20x=21y,$ and $c$ can be any value not equal to $0$.2010-11-15
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    Okay, thanks for your help Derek.2010-11-15