# Deriving possible values of a,b, and c from a system of equations

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 $$(a, b, c)$$.

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

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 $$2$$.