1. **Solve the system of equations:** Given: $$7x + \frac{5y}{8} = 26$$ $$-6x - \frac{3y}{3} = -\frac{1}{3}$$ 2. Simplify the second equation: $$-6x - y = -\frac{1}{3}$$ 3. Multiply the first equation by 8 to clear the denominator: $$8 \times \left(7x + \frac{5y}{8}\right) = 8 \times 26$$ $$56x + 5y = 208$$ 4. Now we have the system: $$56x + 5y = 208$$ $$-6x - y = -\frac{1}{3}$$ 5. Solve the second equation for $y$: $$-6x - y = -\frac{1}{3} \implies y = -6x + \frac{1}{3}$$ 6. Substitute $y$ into the first equation: $$56x + 5\left(-6x + \frac{1}{3}\right) = 208$$ $$56x - 30x + \frac{5}{3} = 208$$ $$26x + \frac{5}{3} = 208$$ 7. Multiply both sides by 3 to clear the fraction: $$3 \times 26x + 3 \times \frac{5}{3} = 3 \times 208$$ $$78x + 5 = 624$$ 8. Subtract 5 from both sides: $$78x = 619$$ 9. Solve for $x$: $$x = \frac{619}{78}$$ 10. Substitute $x$ back into $y = -6x + \frac{1}{3}$: $$y = -6 \times \frac{619}{78} + \frac{1}{3} = -\frac{3714}{78} + \frac{1}{3}$$ 11. Convert $\frac{1}{3}$ to have denominator 78: $$\frac{1}{3} = \frac{26}{78}$$ 12. Calculate $y$: $$y = -\frac{3714}{78} + \frac{26}{78} = -\frac{3688}{78} = -\frac{1844}{39}$$ --- 13. **Solve the equation:** $$\frac{1}{11}(112 - 31x) = \frac{1}{7}(67 + 14x)$$ 14. Multiply both sides by 77 (LCM of 11 and 7): $$7(112 - 31x) = 11(67 + 14x)$$ 15. Expand both sides: $$784 - 217x = 737 + 154x$$ 16. Bring variables to one side and constants to the other: $$784 - 737 = 154x + 217x$$ $$47 = 371x$$ 17. Solve for $x$: $$x = \frac{47}{371}$$ --- 18. **Simplify expressions:** (a) $$9(3x + 2y)^2 - 8(5x - 7y)^2$$ 19. Expand each square: $$(3x + 2y)^2 = 9x^2 + 12xy + 4y^2$$ $$(5x - 7y)^2 = 25x^2 - 70xy + 49y^2$$ 20. Multiply by coefficients: $$9(9x^2 + 12xy + 4y^2) = 81x^2 + 108xy + 36y^2$$ $$-8(25x^2 - 70xy + 49y^2) = -200x^2 + 560xy - 392y^2$$ 21. Combine like terms: $$81x^2 - 200x^2 + 108xy + 560xy + 36y^2 - 392y^2$$ $$= -119x^2 + 668xy - 356y^2$$ (b) $$2x^6 - 28x^5 - 64x^4$$ 22. Factor out the greatest common factor $2x^4$: $$2x^4(x^2 - 14x - 32)$$ 23. Factor the quadratic inside parentheses: $$x^2 - 14x - 32 = (x - 16)(x + 2)$$ 24. Final factorization: $$2x^4(x - 16)(x + 2)$$ --- **Final answers:** - System solution: $$x = \frac{619}{78}, y = -\frac{1844}{39}$$ - Equation solution: $$x = \frac{47}{371}$$ - Expression (a): $$-119x^2 + 668xy - 356y^2$$ - Expression (b): $$2x^4(x - 16)(x + 2)$$