To solve the given system of equations using the Gauss-Seidel method, we need to iteratively approximate the values of $x$, $y$, and $z$.

The system of equations is:

1. $5x + y - z = 10$
2. $2x + 4y + z = 14$
3. $x + y + 8z = 20$

To apply the Gauss-Seidel method, we first rewrite each equation to solve for the primary variable:

1. $x = \frac{10 - y + z}{5}$
2. $y = \frac{14 - 2x - z}{4}$
3. $z = \frac{20 - x - y}{8}$

Let's perform the iterations starting with initial guesses $x^{(0)} = 0$, $y^{(0)} = 0$, $z^{(0)} = 0$, updating each variable in sequence:

First Iteration

Using initial guesses $x^{(0)} = 0$, $y^{(0)} = 0$, $z^{(0)} = 0$:

1. Update $x$: $x^{(1)} = \frac{10 - 0 + 0}{5} = 2.0$

2. Update $y$ using $x^{(1)}$: $y^{(1)} = \frac{14 - 2(2.0) - 0}{4} = \frac{10}{4} = 2.5$

3. Update $z$ using $x^{(1)}$ and $y^{(1)}$: $z^{(1)} = \frac{20 - 2.0 - 2.5}{8} = \frac{15.5}{8} = 1.9375$

Second Iteration

Using updated values $x^{(1)} = 2.0$, $y^{(1)} = 2.5$, $z^{(1)} = 1.9375$:

1. Update $x$: $x^{(2)} = \frac{10 - 2.5 + 1.9375}{5} = \frac{9.4375}{5} = 1.8875$

2. Update $y$ using $x^{(2)}$: $y^{(2)} = \frac{14 - 2(1.8875) - 1.9375}{4} = \frac{8.2875}{4} = 2.071875$

3. Update $z$ using $x^{(2)}$ and $y^{(2)}$: $z^{(2)} = \frac{20 - 1.8875 - 2.071875}{8} = \frac{16.040625}{8} = 2.005078125$

Third Iteration

Using updated values $x^{(2)} = 1.8875$, $y^{(2)} = 2.071875$, $z^{(2)} = 2.005078125$:

1. Update $x$: $x^{(3)} = \frac{10 - 2.071875 + 2.005078125}{5} = \frac{9.933203125}{5} = 1.986640625$

2. Update $y$ using $x^{(3)}$: $y^{(3)} = \frac{14 - 2(1.986640625) - 2.005078125}{4} = \frac{8.021640625}{4} = 1.987109375$

3. Update $z$ using $x^{(3)}$ and $y^{(3)}$: $z^{(3)} = \frac{20 - 1.986640625 - 1.987109375}{8} = \frac{16.02625}{8} = 2.00328125$

After three iterations, the values are approximately $x \approx 1.9866$, $y \approx 1.9871$, and $z \approx 2.0033$. The solution $(x, y, z) = (2, 2, 2)$ provided seems to be an intended target for illustration or may reflect further iterations that continue refining these estimates closer to integers.