Question 334073
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Let *[tex \Large x] represent the amount of tax paid to the state.


Let *[tex \Large y] represent the amount of tax paid to the federal government.


According to the problem, we start with $154K, subtract the amount paid to the feds and then take 10%, like this:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ 0.10(154,000\ -\ y)]


Likewise, the other equation is set up like this:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ 0.30(154,000\ -\ x)]


This system of equations is set up perfectly to use the substitution method, so take the expression that is equal to *[tex \Large x] in the first equation, and put it in place of *[tex \Large x] in the second equation:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ 0.30(154,000\ -\ 0.10(154,000\ -\ y))]


A little arithmetic and a very little algebra gets us to:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ -\ 0.03y\ =\ 41,580]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ \frac{41580}{0.97}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ \approx\ $42,866]


rounded to the nearest dollar.


Make the opposite substitution:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ 0.10(154,000\ -\ 0.30(154,000\ -\ x))]


You can finish up yourself, I think.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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