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In a math problem I arrive at the following system of equations but struggle to solve any variable (in terms of the other two): $$ 2ab + c - 2c^2 = 0 \\ 2bc + a - 2a^2 = 0 \\ 2ac + b - 2b^2 = 0 \\ a + b + c = 2 $$

Please kindly advise any next step, possibly involving algebraic manipulation of any two equations, to find possible values of $(a, b, c)$.

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  • $\begingroup$ Negate the first three and add them. You should find the sum of three squares. $\endgroup$ Nov 19, 2021 at 15:39
  • $\begingroup$ Subtract the first two equation and factor out $(a-c)$. $\endgroup$
    – dxiv
    Nov 20, 2021 at 3:07

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The issue here is that these four equations in three variables are not consistent with each other so there is no solution for $a, b$ and $c$.

For example, if you take three of these equations (which should be sufficient to find solutions since there are three unknowns) $$2ab+c−2c^2=0$$ $$2bc+a−2a^2=0$$and $$a+b+c=2$$

Then you get either $$(a,b,c)=(0,2,0)$$or $$(a,b,c)=(\frac56, \frac13,\frac56)$$ But both of these sets are inconsistent with the equation previously omitted, i.e. $$2ac+b-2b^2=0$$ which no longer holds for either set.

Similarly, if you start with the first three equations, it is not possible for $a+b+c$ to take the value $2$.

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