How to solve this equations $x+y+z=1$
$ya+zc=0.5$
$yb+zd=0.5$
$zc=\frac{1}{3}$
$ya^2+zc^2=\frac{1}{3}$
$yab+zdc=\frac{1}{3}$
$yb^2+zd^2=\frac{1}{3}$
At the moment i got $a=b,c=d,a+2c=2,ya=\frac{1}{6},zc=\frac{1}{3}$
but still stuck

Your solution is almost correct. If you substitute your expressions into your equations, you will see that also $x$ is determined. Finally you can choose, how to express the solution. One possibility is to express every veriable in terms of $d\neq 0,1$, i.e.,
\begin{align*}
x & = \frac{12d^2 - 15d + 4}{12d(d - 1)}, \\
y & = \frac{- 1}{12(d - 1)}, \\
z & = \frac{1}{3d},\\
a & =  - 2(d - 1), \\
b & =  - 2(d - 1),\\
c & = d.
\end{align*}
For $d=1$ and $d=0$ there is no solution.
