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I have 3 unknowns: $\overline{P}_{11}$, $\overline{P}_{12}$, & $\overline{P}_{22}$

And 3 equations...

$$\frac{1}{r}\overline{P}_{12}^2+2b\overline{P}_{12}-2=0$$ $$\frac{1}{r}\overline{P}_{12}\overline{P}_{22}+b\overline{P}_{22}+c\overline{P}_{12}-\overline{P}_{11}-3=0$$ $$\frac{1}{r}\overline{P}_{22}^2+2c\overline{P}_{22}-2\overline{P}_{12}-5=0$$ where $b$, $c$ & $r$ are constants

How do I solve for this system of equations?

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1 Answer

up vote 1 down vote accepted

The first equation is quadratic in $\bar{P}_{12}$, and contains no other unknown. You can thus simply solve it using whatever method you prefer for quadratic equations (completing the square or using the formula).

The last equation is quadratic in $\bar{P}_{22}$, and contains no other unknown except $\bar{P}_{12}$. Thus, once you've found $\bar{P}_{12}$ you can find $\bar{P}_{22}$, also by solving a quadratic equations.

Finally, use the 2nd equation to solve for $\bar{P}_{11}$. It's linear in $\bar{P}_{11}$, so doing that should be very easy.

(And btw, where did $\bar{P}_{21}$ go?)

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Ah, I see that now! Thanks for your help. The $\bar{P}$ matrix is as follows: $$\[ \begin{matrix} \bar{P}_{11} & \bar{P}_{12}\\ \bar{P}_{12} & \bar{P}_{22} \end{matrix} \]$$ –  Kashif Oct 23 '12 at 23:16
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