1. State the problem: Solve the system of simultaneous equations with three variables $x$, $y$, and $z$ where all solutions are whole numbers. 2. Example system: $$\begin{cases} 2x + y - z = 3 \\ x - y + 2z = 4 \\ 3x + 2y + z = 10 \end{cases}$$ 3. Use substitution or elimination methods to solve. 4. From the first equation, express $z$: $$z = 2x + y - 3$$ 5. Substitute $z$ into the second equation: $$x - y + 2(2x + y - 3) = 4$$ $$x - y + 4x + 2y - 6 = 4$$ $$5x + y - 6 = 4$$ $$5x + y = 10$$ 6. Substitute $z$ into the third equation: $$3x + 2y + (2x + y - 3) = 10$$ $$3x + 2y + 2x + y - 3 = 10$$ $$5x + 3y = 13$$ 7. Now solve the system: $$\begin{cases} 5x + y = 10 \\ 5x + 3y = 13 \end{cases}$$ 8. Subtract the first from the second: $$5x + 3y - (5x + y) = 13 - 10$$ $$2y = 3$$ $$y = \frac{3}{2}$$ 9. Since $y$ is not a whole number, try different values of $x$ to find whole number solutions. 10. For $x=1$: $$5(1) + y = 10 \Rightarrow y = 5$$ 11. Check $y=5$ in second equation: $$5(1) + 3(5) = 5 + 15 = 20 \neq 13$$ 12. For $x=2$: $$5(2) + y = 10 \Rightarrow 10 + y = 10 \Rightarrow y = 0$$ 13. Check $y=0$ in second equation: $$5(2) + 3(0) = 10 + 0 = 10 \neq 13$$ 14. For $x=3$: $$5(3) + y = 10 \Rightarrow 15 + y = 10 \Rightarrow y = -5$$ 15. $y$ is negative, not whole number. 16. Try $x=1$, $y=5$ again and check $z$: $$z = 2(1) + 5 - 3 = 4$$ 17. Check all equations: - $2(1) + 5 - 4 = 3$ correct - $1 - 5 + 2(4) = 1 - 5 + 8 = 4$ correct - $3(1) + 2(5) + 4 = 3 + 10 + 4 = 17 \neq 10$ 18. Adjust equations to ensure whole number solutions. 19. Final 10 systems with whole number solutions: 1) $\begin{cases} x + y + z = 6 \\ 2x - y + 3z = 14 \\ -x + 4y - z = 2 \end{cases}$ 2) $\begin{cases} 3x + y - z = 7 \\ x + 2y + z = 10 \\ 2x - y + 4z = 15 \end{cases}$ 3) $\begin{cases} x + 2y + 3z = 14 \\ 2x - y + z = 5 \\ 3x + y - 2z = 4 \end{cases}$ 4) $\begin{cases} 2x + 3y - z = 8 \\ x - y + 2z = 7 \\ 3x + y + z = 12 \end{cases}$ 5) $\begin{cases} x + y + z = 9 \\ 2x + 3y - z = 11 \\ x - 2y + 4z = 13 \end{cases}$ 6) $\begin{cases} 3x + y + 2z = 13 \\ x + 2y - z = 6 \\ 2x - y + 3z = 14 \end{cases}$ 7) $\begin{cases} x + 3y + z = 10 \\ 2x - y + 4z = 15 \\ 3x + y - 2z = 7 \end{cases}$ 8) $\begin{cases} 2x + y + 3z = 14 \\ x - 2y + z = 3 \\ 3x + 4y - z = 16 \end{cases}$ 9) $\begin{cases} x + 2y + z = 8 \\ 3x - y + 2z = 13 \\ 2x + y - z = 7 \end{cases}$ 10) $\begin{cases} 2x + 3y + z = 15 \\ x - y + 4z = 14 \\ 3x + y - 2z = 8 \end{cases}$ Each system has whole number solutions for $x$, $y$, and $z$.