Solving a system of polynomial equations

Question

How can I solve a system of polynomial equations like this one

Maybe I'm missing a very basic trick... Can anybody suggest me an approach?

Answer 1

Take the reciprocals of each and multiply by $abcdef$ to obtain the equations $$a^2 = \frac14 abcdef \\ b^2 = \frac19 abcdef \\ \vdots$$

Multiply all original equations to get $$(abcdef)^4 = 4\cdot 9\cdot 16 \cdots = 1$$

Now you should be able to tell the solution without using a calculator.

Answer 2

Take a logarithm and solve a linear system

\begin{align} \left[\begin{matrix} -1 & 1 & 1 & 1 & 1& 1 \\ 1 & -1 & 1 & 1 & 1& 1 \\ 1 & 1 & -1 & 1 & 1& 1 \\ 1 & 1 & 1 & -1 & 1& 1 \\ 1 & 1 & 1 & 1 & -1& 1 \\ 1 & 1 & 1 & 1 & 1&-1 \end{matrix}\right] \left[\begin{matrix} \ln a \\ \ln b \\ \ln c \\ \ln d \\ \ln e \\ \ln f \end{matrix}\right] &=\left[\begin{matrix} \ln 4 \\ \ln 9 \\ \ln 16 \\ -\ln 4 \\ -\ln 9 \\ -\ln 16\end{matrix} \right] \end{align}

\begin{align} \left[\begin{matrix} \ln a \\ \ln b \\ \ln c \\ \ln d \\ \ln e \\ \ln f \end{matrix}\right] = \tfrac18 \left[\begin{matrix} -3 & 1 & 1 & 1 & 1& 1 \\ 1 & -3 & 1 & 1 & 1& 1 \\ 1 & 1 & -3 & 1 & 1& 1 \\ 1 & 1 & 1 & -3 & 1& 1 \\ 1 & 1 & 1 & 1 & -3& 1 \\ 1 & 1 & 1 & 1 & 1&-3 \end{matrix}\right] {} \left[\begin{matrix} \ln 4 \\ \ln 9 \\ \ln 16 \\ -\ln 4 \\ -\ln 9 \\ -\ln 16 \end{matrix}\right] \end{align}