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May 9th, 2018, 02:20 AM   #1
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System of equations

$\begin{aligned}
x^2+xy+y^2=1\\
x^2+xz+z^2=4\\
y^2+zy+z^2=7
\end{aligned}$
Subtracting 1st from 2nd, and 2nd from 3rd:
$\begin{aligned}
&(z-y)(x+y+z)=3\\
&(y-x)(x+y+z)=3\\
&\implies z-y=y-x\Leftrightarrow z=2y-x
\end{aligned}$
How do I proceed from here? I've been at it for quite a while, and I still feel like I'm just short of untangling it. Thanks.
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May 9th, 2018, 02:44 AM   #2
lua
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Try to do this:

$\displaystyle (y-x) \cdot 3y=3$, that is:

$\displaystyle (y-x) \cdot y=1$.

From here: $\displaystyle x=y-\frac{1}{y}$.

Put it in the first equation, and you get:

$\displaystyle 3y^{2}+\frac{1}{y^{2}}=4$

Substitute $\displaystyle t=y^{2}$ and you have:

$\displaystyle 3t^{2}-4t+1=0$
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