Is there a system of linear equations with 3 variables such that its solution set is: $\{(a,b,c)|a^2=b \}$?
It's enough to show that for one equation: $Ax+By+Cz=D$ the solution set doesn't work.
Take $(0,0,c)$ for all $c$ we have $Cc=D$ which can hold only if $C=D=0$.
So we have a homogeneous equation: $Ax+By=0$ and from placing the solutions $(1,1,0),(-1,1,0)$ we get that $A=B=0$.
Now I'm not sure how this contradicts the solution set.
The solution set of such a system has to be a plane in 3 dimensions. So you cannot have a solution set such as you describe.