Can I divide entire equations by each other when trying to solve a system of non-linear equations?

Question

I have the following equations:

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1. $\frac{F}{(W-\phi)t}=\sigma_{_L}$
2. $\frac{F}{(W^2-\phi^2)t}=\tau_{_L}$
3. $\frac{F}{\phi t}=\sigma_{_L}$
4. $\frac{F}{\phi^2\pi/4}=\tau_{_P}$

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I then solve for F

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1. $F=(W-\phi)t\sigma_{_L}$
2. $F=(W^2-\phi^2)t\tau_{_L}$
3. $F=\phi t\sigma_{_L}$
4. $F=\phi^2\frac{\pi}{4}\tau_{_P}$

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Since they are all in terms of F, I can divide any of these equations by another and I should get 1.

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1./2. $\frac{(W-\phi)t\sigma_{_L}}{(W^2-\phi^2)t\tau_{_L}}=1$

3./4. $\frac{\phi t\sigma_{_L}}{\phi^2\frac{\pi}{4}\tau_{_P}}=1$

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Now I can set them equal to each other and solve for different variables.

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1./2. = 3./4. $\frac{(W-\phi)t\sigma_{_L}}{(W^2-\phi^2)t\tau_{_L}}=\frac{\phi t\sigma_{_L}}{\phi^2\frac{\pi}{4}\tau_{_P}}$

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This seems like a logical approach, but I am not sure if there is a flaw in my logic since I could end up with a very different equation (after simplification) if I instead did 1./4. = 2./3. or any permutation thereof.

Is my approach incorrect? Is there a better way to do this (besides doing substitution)?