1. **State the problem:** Solve the system of linear equations using Gaussian elimination: $$\begin{cases}-2x - 3y + 2z + u = 0 \\ -y + z + y = 1 \\ -4x + 2y - u = 0 \end{cases}$$ 2. **Write the augmented matrix:** $$\left[\begin{array}{cccc|c}-2 & -3 & 2 & 1 & 0 \\ 0 & -1 & 0 & 1 & 1 \\ -4 & 2 & 0 & -1 & 0 \end{array}\right]$$ 3. **Simplify the second equation:** Note that $-y + z + y = z$, so the second equation is $z = 1$. 4. **Use the second row to express $z$: $z = 1$.** Substitute $z=1$ into the first and third rows. 5. **Substitute $z=1$ into row 1:** $$-2x - 3y + 2(1) + u = 0 \implies -2x - 3y + 2 + u = 0$$ Simplify: $$-2x - 3y + u = -2$$ 6. **Substitute $z=1$ into row 3:** $$-4x + 2y - u = 0$$ 7. **Rewrite the system with two equations and variables $x,y,u$: ** $$\begin{cases}-2x - 3y + u = -2 \\ -4x + 2y - u = 0 \end{cases}$$ 8. **Add the two equations to eliminate $u$: ** $$(-2x - 3y + u) + (-4x + 2y - u) = -2 + 0$$ Simplify: $$-6x - y = -2$$ 9. **Express $y$ in terms of $x$: ** $$-6x - y = -2 \implies y = -6x + 2$$ 10. **Use the first equation to express $u$: ** $$-2x - 3y + u = -2 \implies u = -2 + 2x + 3y$$ Substitute $y = -6x + 2$: $$u = -2 + 2x + 3(-6x + 2) = -2 + 2x - 18x + 6 = 4 - 16x$$ 11. **Summary of solutions:** $$z = 1, \quad y = -6x + 2, \quad u = 4 - 16x, \quad x = x \text{ (free parameter)}$$ This is the parametric solution of the system. **Final answer:** $$\boxed{\left(x, y, z, u\right) = \left(x, -6x + 2, 1, 4 - 16x\right), \quad x \in \mathbb{R}}$$