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tldr; Please help me solve for x in the following equation:

$$\frac{tmp -\frac{ptr\cdot tv\cdot x}{12}}{\frac{ir}{12} / (1- (1+ \frac{ir}{12})^{-360})} =x$$

I'm trying to create a formula that will allow me to calculate the largest mortgage I can afford (x) based on a monthly budget given the following parameters:

tmp - Total monthly payment
ptr - Property tax rate
tv  - Taxable value (percentage of home value that can be taxed)
ir  - Interest rate of mortgage

I started with this formula:

$$\frac{tmp}{\frac{ir}{12} / (1- (1+ \frac{ir}{12}))^{-360})} = x$$

which works well, but it doesn't take into consideration property taxes. In city A 1000 a month might get me a 300k house but in city B, the property taxes are higher and I can only get a 280k house for the same monthly cost.

So to consider the property tax as a mandatory part of the monthly payment, I replaced $tmp$ with $tmp - \frac{ptr\cdot tv\cdot x}{12}$. I think this will work, but I need to get that pesky $x$ out of there.

So, given the final formula, how do I solve for x here?

$$\frac{tmp -\frac{ptr\cdot tv\cdot x}{12}}{\frac{ir}{12} / (1- (1+ \frac{ir}{12})^{-360})} =x$$

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    Welcome to Math.SE. Please format your equations with MathJax. [tutorial](http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference) – 2017-01-04
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    Thanks for pointing that out. I updated the formatting and think I got everything right. – 2017-01-04

1 Answers 1

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$${tpm-{ptr \cdot tv \cdot x\over12}\over{{ir\over12}\over(1-(1+{ir\over12})^{-360})}}=x\longrightarrow tpm-{ptr \cdot tv \cdot x\over12}={{ir\over12}\over(1-(1+{ir\over12})^{-360})} \cdot x$$ $$tpm={ptr \cdot tv \cdot x\over12}+{{ir\over12}\over(1-(1+{ir\over12})^{-360})}x\longrightarrow x\left[{ptr \cdot tv\over12}+{{ir\over12}\over(1-(1+{ir\over12})^{-360})}\right]=tpm$$ $$x={tpm\over\left[{ptr \cdot tv\over12}+{{ir\over12}\over(1-(1+{ir\over12})^{-360})}\right]}={tpm\over\left[{ptr \cdot tv\over12}+{ir\over12(1-(1+{ir\over12})^{-360})}\right]}=$$

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    Thanks. This indeed is the correct answer. Unfortunately my original formula is off somewhere and I'm not getting the correct result. I should be able to take it from here though. – 2017-01-04