Is it possible to find solutions of this system expressed with $a$:
$x^2+y^2+xy=a$
$x^2+z^2+zx=a$
$y^2+z^2-yz=a$
Thanks for help.
System of equations with with 3 variables
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algebra-precalculus
systems-of-equations
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2What are $(2)-(1)$ and $(3)-(1)$? – 2017-02-25
1 Answers
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It is not always possible, at least not over the real numbers, to find solutions of this system. We obtain either $y^2=a$ or $x^2=a$, which has both no real solution for $a<0$. We argue as follows: $(2)-(1)$ gives $$ (z-y)(x+y+z)=0. $$ For $z=y$ we obtain $y^2=a$ from $(3)$, and this has no solution for $a<0$. For $a\ge 0$ it has a solution, namely $y^2=a$, $x(x+y)=0$.
In the other case, $y\neq z$ we obtain the solution $z=-x-y$, and $y(x+y)=0$, $a=x^2$.