1. **State the problem:** Solve the system of equations: $$\begin{cases} w+x+y+z=2 \\ 2w - x - y + 2z = 7 \\ 2w + 3x + 2y - z = -2 \\ 3w - 2x - y - 3z = -2 \end{cases}$$ 2. **Write the system in matrix form:** $$\begin{bmatrix} 1 & 1 & 1 & 1 \\ 2 & -1 & -1 & 2 \\ 2 & 3 & 2 & -1 \\ 3 & -2 & -1 & -3 \end{bmatrix} \begin{bmatrix} w \\ x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 2 \\ 7 \\ -2 \\ -2 \end{bmatrix}$$ 3. **Use elimination or substitution to solve:** - From equation 1: $$w = 2 - x - y - z$$ - Substitute into equations 2, 3, and 4: Equation 2: $$2(2 - x - y - z) - x - y + 2z = 7$$ $$4 - 2x - 2y - 2z - x - y + 2z = 7$$ $$4 - 3x - 3y = 7$$ $$-3x - 3y = 3$$ $$x + y = -1$$ Equation 3: $$2(2 - x - y - z) + 3x + 2y - z = -2$$ $$4 - 2x - 2y - 2z + 3x + 2y - z = -2$$ $$4 + x - 3z = -2$$ $$x - 3z = -6$$ Equation 4: $$3(2 - x - y - z) - 2x - y - 3z = -2$$ $$6 - 3x - 3y - 3z - 2x - y - 3z = -2$$ $$6 - 5x - 4y - 6z = -2$$ $$-5x - 4y - 6z = -8$$ 4. **Use $x + y = -1$ to express $y = -1 - x$ and substitute into the other equations:** Equation $x - 3z = -6$ remains the same. Equation 4: $$-5x - 4(-1 - x) - 6z = -8$$ $$-5x + 4 + 4x - 6z = -8$$ $$-x + 4 - 6z = -8$$ $$-x - 6z = -12$$ $$x + 6z = 12$$ 5. **Now solve the system:** $$\begin{cases} x - 3z = -6 \\ x + 6z = 12 \end{cases}$$ Subtract the first from the second: $$(x + 6z) - (x - 3z) = 12 - (-6)$$ $$x + 6z - x + 3z = 18$$ $$9z = 18$$ $$z = 2$$ 6. **Find $x$ using $x - 3z = -6$:** $$x - 3(2) = -6$$ $$x - 6 = -6$$ $$x = 0$$ 7. **Find $y$ using $y = -1 - x$:** $$y = -1 - 0 = -1$$ 8. **Find $w$ using $w = 2 - x - y - z$:** $$w = 2 - 0 - (-1) - 2 = 2 + 1 - 2 = 1$$ **Final solution:** $$\boxed{w=1, x=0, y=-1, z=2}$$