1. **State the problems:** We have several systems and inequalities to analyze and solve. --- ### System 1: Solve for $x$, $y$, and $z$: $$\begin{cases} 5y - 4z = 12 \\ 3x - 4y + 2z = 8 \\ -6x + 3z = 15 \end{cases}$$ 2. From the third equation: $$-6x + 3z = 15 \implies -6x = 15 - 3z \implies x = \frac{3z - 15}{6} = \frac{z}{2} - \frac{5}{2}$$ 3. Substitute $x$ into the second equation: $$3\left(\frac{z}{2} - \frac{5}{2}\right) - 4y + 2z = 8$$ $$\Rightarrow \frac{3z}{2} - \frac{15}{2} - 4y + 2z = 8$$ $$\Rightarrow \frac{3z}{2} + 2z - 4y = 8 + \frac{15}{2}$$ $$\Rightarrow \frac{7z}{2} - 4y = \frac{16}{2} + \frac{15}{2} = \frac{31}{2}$$ 4. Multiply whole equation by 2: $$7z - 8y = 31$$ 5. From the first equation: $$5y - 4z = 12$$ 6. Solve the system: $$\begin{cases} 5y - 4z = 12 \\ 7z - 8y = 31/2 \end{cases}$$ Multiply first equation by 8: $$40y - 32z = 96$$ Multiply second equation by 5: $$35z - 40y = 77.5$$ 7. Add these equations: $$40y - 32z + 35z - 40y = 96 + 77.5 \implies 3z = 173.5 \implies z = \frac{173.5}{3} = 57.8333...$$ 8. Plug $z$ back into $5y - 4z = 12$: $$5y - 4(57.8333) = 12 \implies 5y = 12 + 231.3333 = 243.3333 \implies y = 48.6667$$ 9. Calculate $x$: $$x = \frac{z}{2} - \frac{5}{2} = \frac{57.8333}{2} - 2.5 = 28.9167 - 2.5 = 26.4167$$ --- ### Inequalities: Solve for $x$ in each: 10. $$| -2x + 7 | \geq 2$$ $$\Rightarrow -2x + 7 \geq 2 \quad \text{or} \quad -2x + 7 \leq -2$$ $$-2x \geq -5 \implies x \leq 2.5$$ $$-2x \leq -9 \implies x \geq 4.5$$ Solution: $$x \leq 2.5\;\text{or}\;x \geq 4.5$$ 11. $$| 3x - 7 | - 5 \leq 12 \Rightarrow | 3x - 7 | \leq 17$$ $$-17 \leq 3x - 7 \leq 17$$ $$-10 \leq 3x \leq 24$$ $$-\frac{10}{3} \leq x \leq 8$$ 12. $$| 2x - 3 | \leq 6$$ $$-6 \leq 2x - 3 \leq 6$$ $$-3 \leq 2x \leq 9$$ $$-\frac{3}{2} \leq x \leq \frac{9}{2} = 4.5$$ --- ### System 2: Solve $$\begin{cases} x + 2y - 3z = 3 \\ 3x - 3y - z = 6 \\ 2y - z = 9 \end{cases}$$ 13. From third equation: $$z = 2y - 9$$ 14. Substitute $z$ into first and second equations: $$x + 2y - 3(2y - 9) = 3 \implies x + 2y - 6y + 27 = 3 \implies x - 4y = -24$$ $$3x - 3y - (2y - 9) = 6 \implies 3x - 3y - 2y + 9 = 6 \implies 3x - 5y = -3$$ 15. Solve system: $$\begin{cases} x - 4y = -24 \\ 3x - 5y = -3 \end{cases}$$ Multiply first by 3: $$3x - 12y = -72$$ Subtract second: $$(3x - 12y) - (3x - 5y) = -72 - (-3) \implies -7y = -69 \implies y = 9.8571$$ 16. Find $x$: $$x - 4(9.8571) = -24 \implies x = -24 + 39.4284 = 15.4284$$ 17. Find $z$: $$z = 2(9.8571) - 9 = 19.7142 - 9 = 10.7142$$ --- ### Optimization problem: Minimize $$Y = 2x_1 + 5x_2$$ Subject to $$12x_1 + 10x_2 \leq 125$$ $$6x_1 + x_2 \geq 56$$ $$x_1, x_2 \geq 0$$ 18. The feasible region is bounded by: $$12x_1 + 10x_2 \leq 125$$ $$6x_1 + x_2 \geq 56$$ Consider equality lines: $$12x_1 + 10x_2 = 125$$ $$6x_1 + x_2 = 56$$ 19. Solve second for $x_2$: $$x_2 = 56 - 6x_1$$ Substitute into first: $$12x_1 + 10(56 - 6x_1) = 125$$ $$12x_1 + 560 - 60x_1 = 125$$ $$-48x_1 = -435 \implies x_1 = \frac{435}{48} = 9.0625$$ 20. Calculate $x_2$: $$x_2 = 56 - 6(9.0625) = 56 - 54.375 = 1.625$$ 21. Evaluate $Y$ at this point: $$Y = 2(9.0625) + 5(1.625) = 18.125 + 8.125 = 26.25$$ 22. Also check vertices where constraints intersect axes (and satisfy inequalities) for minimal $Y$: - When $x_1=0$, from $6(0)+x_2 \geq 56 \Rightarrow x_2 \geq 56$ but this violates $12x_1 + 10x_2 \leq 125$ because $10\times 56=560 >125$. - When $x_2=0$, $6x_1 \geq 56 \Rightarrow x_1 \geq 9.3333$ and $12x_1\leq 125$, $x_1=9.3333$ works: $$Y = 2(9.3333)+5(0) = 18.6667$$ 23. Check $Y$ at $x_1=9.3333, x_2=0$ (possible feasible point): Constraint 1: $$12(9.3333) + 10(0) = 112 \leq 125$$ Constraint 2: $$6(9.3333)+0 = 56 \geq 56$$ This point is feasible and $$Y=18.6667 < 26.25$$ **Minimum $Y$ is approximately $18.67$ at $x_1=9.3333$, $x_2=0$.**