1. Given the system of equations: $$x + y - z = 2$$ $$2x - 2y - z = -1$$ $$3x + 2y + z = 8$$ We want to find the values of $x$, $y$, and $z$. 2. Add the first and second equations to eliminate $z$: $$(x + y - z) + (2x - 2y - z) = 2 + (-1)$$ $$3x - y - 2z = 1$$ 3. Use the third equation: $$3x + 2y + z = 8$$ 4. From the first equation, express $z$: $$z = x + y - 2$$ 5. Substitute $z$ into the second and third equations: Second equation: $$2x - 2y - (x + y - 2) = -1$$ $$2x - 2y - x - y + 2 = -1$$ $$x - 3y = -3$$ Third equation: $$3x + 2y + (x + y - 2) = 8$$ $$3x + 2y + x + y - 2 = 8$$ $$4x + 3y = 10$$ 6. Solve the system: $$\begin{cases} x - 3y = -3 \\ 4x + 3y = 10 \end{cases}$$ Add the two equations: $$(x - 3y) + (4x + 3y) = -3 + 10$$ $$5x = 7$$ $$x = \frac{7}{5}$$ 7. Substitute $x$ back to find $y$: $$\frac{7}{5} - 3y = -3$$ $$-3y = -3 - \frac{7}{5} = -\frac{15}{5} - \frac{7}{5} = -\frac{22}{5}$$ $$y = \frac{22}{15}$$ 8. Find $z$: $$z = x + y - 2 = \frac{7}{5} + \frac{22}{15} - 2 = \frac{21}{15} + \frac{22}{15} - \frac{30}{15} = \frac{13}{15}$$ 9. Calculate $12xyz$: $$12 \times \frac{7}{5} \times \frac{22}{15} \times \frac{13}{15} = 12 \times \frac{7 \times 22 \times 13}{5 \times 15 \times 15} = 12 \times \frac{2002}{1125} = \frac{24024}{1125} = \frac{32032}{1500} = \frac{24024}{1125} \approx 21.36$$ 10. For the parabola passing through points $(7,-1)$, $(2,-1)$, and $(0,-1)$, assume the form: $$y = ax^2 + bx + c$$ Substitute points: $$-1 = 49a + 7b + c$$ $$-1 = 4a + 2b + c$$ $$-1 = c$$ 11. From the third equation, $c = -1$. Substitute $c$ into the first two: $$49a + 7b - 1 = -1 \Rightarrow 49a + 7b = 0$$ $$4a + 2b - 1 = -1 \Rightarrow 4a + 2b = 0$$ 12. Simplify: $$7a + b = 0$$ $$2a + b = 0$$ Subtract second from first: $$(7a + b) - (2a + b) = 0 - 0$$ $$5a = 0 \Rightarrow a = 0$$ Then $b = 0$. 13. So the parabola is: $$y = -1$$ This is a horizontal line, not matching options A, B, or C. 14. For the parabola $y = (1-5), (2,6)$ is unclear; assuming a typo, no solution can be given. Final answers: - Solution to system: $x=\frac{7}{5}$, $y=\frac{22}{15}$, $z=\frac{13}{15}$ - $12xyz \approx 21.36$ - Parabola through given points is $y = -1$