1. **State the problem:** Solve the system of equations with three variables $x$, $y$, and $z$. 2. **General form:** A system of three linear equations can be written as: $$\begin{cases} a_1x + b_1y + c_1z = d_1 \\ a_2x + b_2y + c_2z = d_2 \\ a_3x + b_3y + c_3z = d_3 \end{cases}$$ 3. **Method:** We can use substitution, elimination, or matrix methods (like Gaussian elimination) to solve. 4. **Example:** Suppose the system is: $$\begin{cases} 2x + y - z = 8 \\ -3x - y + 2z = -11 \\ -2x + y + 2z = -3 \end{cases}$$ 5. **Step 1: Eliminate one variable.** Add equation 1 and equation 2: $$ (2x + y - z) + (-3x - y + 2z) = 8 + (-11) $$ $$ \Rightarrow (2x - 3x) + (y - y) + (-z + 2z) = -3 $$ $$ \Rightarrow -x + z = -3 $$ 6. **Step 2: Express $z$ in terms of $x$:** $$ z = -3 + x $$ 7. **Step 3: Substitute $z$ into equation 3:** $$ -2x + y + 2(-3 + x) = -3 $$ $$ -2x + y - 6 + 2x = -3 $$ $$ \cancel{-2x} + y - 6 + \cancel{2x} = -3 $$ $$ y - 6 = -3 $$ $$ y = 3 $$ 8. **Step 4: Substitute $y=3$ and $z = -3 + x$ into equation 1:** $$ 2x + 3 - (-3 + x) = 8 $$ $$ 2x + 3 + 3 - x = 8 $$ $$ (2x - x) + 6 = 8 $$ $$ x + 6 = 8 $$ $$ x = 2 $$ 9. **Step 5: Find $z$:** $$ z = -3 + x = -3 + 2 = -1 $$ 10. **Final answer:** $$ (x, y, z) = (2, 3, -1) $$ This solution satisfies all three equations.