1. **Problem:** Solve the system of linear equations: $$\begin{cases} x - y - 2z = 0 \\ -2x + 6y = 1 \\ -x + 2y + z = 1 \end{cases}$$ 2. **Formula and rules:** To solve a system of linear equations, we can use substitution, elimination, or matrix methods such as Gaussian elimination. Here, we will use substitution/elimination. 3. **Step 1:** From the first equation, express $x$ in terms of $y$ and $z$: $$x = y + 2z$$ 4. **Step 2:** Substitute $x = y + 2z$ into the second equation: $$-2(y + 2z) + 6y = 1$$ Simplify: $$-2y - 4z + 6y = 1$$ $$4y - 4z = 1$$ 5. **Step 3:** Substitute $x = y + 2z$ into the third equation: $$-(y + 2z) + 2y + z = 1$$ Simplify: $$-y - 2z + 2y + z = 1$$ $$y - z = 1$$ 6. **Step 4:** Now we have two equations with two variables: $$\begin{cases} 4y - 4z = 1 \\ y - z = 1 \end{cases}$$ 7. **Step 5:** From the second equation, express $y$: $$y = 1 + z$$ 8. **Step 6:** Substitute $y = 1 + z$ into the first equation: $$4(1 + z) - 4z = 1$$ Simplify: $$4 + 4z - 4z = 1$$ $$4 = 1$$ 9. **Step 7:** The equation $4 = 1$ is a contradiction, meaning the system has no solution. **Final answer:** The system of equations is inconsistent and has no solution.