1. **State the problem:** Solve the system of linear equations: $$\begin{cases} x + y - z = 1 \\ 2x + 2y - 3z = 1 \\ 4x - 2y - z = 1 \end{cases}$$ 2. **Use substitution or elimination method:** We will use elimination to find $x$, $y$, and $z$. 3. **Label the equations:** $$\text{(1)}\quad x + y - z = 1$$ $$\text{(2)}\quad 2x + 2y - 3z = 1$$ $$\text{(3)}\quad 4x - 2y - z = 1$$ 4. **Eliminate variables step-by-step:** - Multiply equation (1) by 2: $$2x + 2y - 2z = 2$$ - Subtract equation (2) from this result: $$ (2x + 2y - 2z) - (2x + 2y - 3z) = 2 - 1 $$ $$ -2z + 3z = 1 $$ $$ z = 1 $$ 5. **Substitute $z=1$ into equation (1):** $$ x + y - 1 = 1 $$ $$ x + y = 2 $$ 6. **Substitute $z=1$ into equation (3):** $$ 4x - 2y - 1 = 1 $$ $$ 4x - 2y = 2 $$ 7. **Solve the system:** $$ \begin{cases} x + y = 2 \\ 4x - 2y = 2 \end{cases} $$ - Multiply the first equation by 2: $$ 2x + 2y = 4 $$ - Add this to the second equation: $$ (4x - 2y) + (2x + 2y) = 2 + 4 $$ $$ 6x = 6 $$ $$ x = 1 $$ 8. **Find $y$ using $x=1$ in $x + y = 2$:** $$ 1 + y = 2 $$ $$ y = 1 $$ 9. **Final solution:** $$ (x, y, z) = (1, 1, 1) $$