Consider the system of equations with constraints $$ \begin{cases} x+y+z+t+u+v=3(a+b), \\ x+y+2(z+t)+3u=6b\\ 0 \leq x,y,z,t,u,v \leq 1, \end{cases} $$ here $0 \leq a,b \leq 1$ are fixed parameters.
I need to find at least one non-trivial solution of the equation. Under nontrivial I mean a solution that is differ from $0$ and $1$, it would be very preferable for almost all $a,b.$ Better if solutions were expresed in terms of $a,b$. If not then must be an algorithm to calculate it.
My attempt. I treated the problem as an optimization problem and try to use the simplex method. Unfortunatelly I get very often a solution with many zeros and ones. For example if $a=0.22, b=0.34$ I get $$ t= 0.52,u= 0.0,v= 0.16,x= 1.0,y= 0.0,z= 0.0$$ and it is not so good.
Any ideas?
Adding and subtracting, we obtain the set of all solutions to be:
$y=6a+u-2v-x$ and $z=-3a+3b-t-2u+v$.
So for example setting $a=\frac{1}{2}, u=\frac{1}{6}, v=\frac{6}{7},x=\frac{1}{2}, t=\frac{1}{2}$ and $b=\frac{2}{3}$ gives $y=\frac{20}{21}$ and $z=\frac{11}{21}$.