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I am trying to find all real solutions to the following set of equations: $\begin{align*} b(a-c)-ad&=0\\ 2ab+cb+ad-2cd&=0 \end{align*}$

My algebra is a bit rusty, and I really have no idea where to begin (aside from maybe a long sequence of 'brute force' by substitution). I would appreciate help with this.

Thanks in advance!

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    Srivatsan: that's exactly the kind of answer I am looking for, yes. Everyone: I'm sorry for so many vague details, not used to posting questions here!2011-09-02

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HINT Your question can be equivalently written as: $ \begin{align*} 3ab=2cd \tag{1} \\ bc+ad = ab \tag{2} \end{align*} $

You can eliminate $d$ from the system by taking $a \times (1) + 2c \times (2)$, when you will get the equation: $ \ldots (\text{do the algebra and find the equation}) $ Collect together all terms on one side and factor the equations. You will find that either $b = 0$ or $\ldots$. (Fill in the blank.)

  • Case 1: Suppose $b = 0$. Can you handle this case?

  • Case 2: Suppose $\ldots$ holds. Then in this case, $a = c = 0$. (Can you prove this? It is slightly nontrivial to show this.) In this case, what can you say about $b$ and $d$?

So, to conclude, what are all the solutions of the equation?

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    @mixedmath Thanks. :-)2011-09-03