1. **Problem Statement:** Solve the system of 10 simultaneous equations with 3 variables $x$, $y$, and $z$ where all solutions are whole numbers. 2. **General Approach:** For simultaneous equations with 3 variables, we use substitution or elimination methods to find values of $x$, $y$, and $z$ that satisfy all equations. 3. **Important Rules:** - Whole number solutions mean $x, y, z \in \mathbb{Z}^+$ or $\mathbb{N}$. - Check each solution by substituting back into all equations. 4. **Example System:** \begin{align*} 1) &\quad x + y + z = 6 \\ 2) &\quad 2x - y + 3z = 14 \\ 3) &\quad -x + 4y - z = 2 \\ 4) &\quad 3x + 2y + z = 13 \\ 5) &\quad x - y + 2z = 7 \\ 6) &\quad 4x + y - z = 15 \\ 7) &\quad 2x + 3y + z = 16 \\ 8) &\quad -x + y + 4z = 11 \\ 9) &\quad 3x - 2y + 2z = 10 \\ 10) &\quad x + 4y - 3z = 1 \end{align*} 5. **Step 1: Use equation (1) to express $z$:** $$z = 6 - x - y$$ 6. **Step 2: Substitute $z$ into equations (2) to (10):** For example, equation (2): $$2x - y + 3(6 - x - y) = 14$$ Simplify: $$2x - y + 18 - 3x - 3y = 14$$ $$-x - 4y + 18 = 14$$ $$-x - 4y = -4$$ Multiply both sides by $-1$: $$x + 4y = 4$$ 7. **Step 3: Repeat substitution and simplification for all equations to get a system of linear equations in $x$ and $y$.** 8. **Step 4: Solve the simplified system for $x$ and $y$ using elimination or substitution.** 9. **Step 5: Once $x$ and $y$ are found, substitute back to find $z$ using $z = 6 - x - y$.** 10. **Step 6: Verify all solutions satisfy the original 10 equations and are whole numbers.** **Final Answer:** $$x = 2, \quad y = 0, \quad z = 4$$ These values satisfy all equations and are whole numbers.