Four, (Full marks 12 points) Given that $a$ and $b$ are the two real roots of the quadratic equation $t^{2}-t-1=0$. Solve the system of equations
$$
\left\{\begin{array}{l}
\frac{x}{a}+\frac{y}{b}=1+x, \\
\frac{x}{b}+\frac{y}{a}=1+y .
\end{array}\right.
$$
Let $a, b$ be the roots of the equation $t^{2}-t-1=0$. By Vieta's formulas, we have $a+b=1, ab=-1$.

Multiplying the left side of equations (1) and (2) by $ab$, and the right side by -1, we get
$$
\left\{\begin{array}{l}
b x + a y = -(1 + x), \\
a x + b y = -(1 + y).
\end{array}\right.
$$
Adding (3) and (4) and simplifying, we get $x + y = -1$.
$y = -(x + 1)$, substituting into (3) and simplifying, we get
$(b - a + 1) x = a - 1$,
$2 b x = -b$,
$x = -\frac{1}{2}$, thus we get $y = -\frac{1}{2}$.