MHB Simple Simultaneous equation help

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The discussion revolves around solving a system of simultaneous equations involving variables x1, x2, and a parameter λ. The user successfully derived λ* as λ* = √(w1w2/ȳ) but struggles to find the correct form of x*. Another participant points out that the correct solution for x* is x* = (√(w2ȳ/w1), √(w1ȳ/w2)), correcting the user's earlier omission of square root signs. The user expresses gratitude for the clarification, indicating that the solution was confirmed. The focus remains on solving the equations accurately within the given constraints.
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Assuming $x_1, x_2 \geq 0, \lambda \neq 0, w_1,w_2 > 0$

We have the equalities:

$$w_1 - \lambda x_2 = 0 ... (1)$$
$$w_2 - \lambda x_1 = 0 ... (2)$$
$$\bar y - x_1x_2= 0 ... (3)$$

My solutions say that $\lambda^* = \sqrt\frac{w_1w_2}{\bar y}$
Which I was able to solve myself.

The other solution is $x^* = (\frac{w_2 \bar y}{w_1}, \frac{w_1 \bar y}{w_2})$

Which I cannot seem to get. Would anyone be so kind as to point out how to obtain solutions for $x^*$ ?

I tried:

$x_1= \frac{w_2}{\lambda}$ and $x_2 = \frac{w_1}{\lambda}$

then put these into the 3rd equation, but ended up gettig
$\bar y = \frac{w_1w_2}{\lambda^2}$ which I couldn't see turning into what I needed either way - even just subbing in one of the x-values at a time, but to no avail.
 
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anyone? :(

If I wasn't clear, $x^* = (x_1,x_2)$
 
Hi nacho,

The square root signs are missing in the answer for $x^*$. It should be $x^* = (\sqrt{w_2 \bar{y}/w_1}, \sqrt{w_1\bar{y}/w_2})$.
 
this was a suspicion of mine - thank you!
 
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