1. Solve the system of linear equations: $$7x + \frac{5y}{8} = 26$$ $$-6x - \frac{3y}{3} = -\frac{1}{3}$$ Step 1: Simplify the second equation: $$-6x - y = -\frac{1}{3}$$ Step 2: Multiply the first equation by 8 to clear the fraction: $$56x + 5y = 208$$ Step 3: Solve the system: From second equation: $$y = -6x + \frac{1}{3}$$ Substitute into first: $$56x + 5(-6x + \frac{1}{3}) = 208$$ $$56x - 30x + \frac{5}{3} = 208$$ $$26x = 208 - \frac{5}{3} = \frac{624}{3} - \frac{5}{3} = \frac{619}{3}$$ $$x = \frac{619}{78}$$ Step 4: Find y: $$y = -6 \times \frac{619}{78} + \frac{1}{3} = -\frac{3714}{78} + \frac{1}{3} = -\frac{3714}{78} + \frac{26}{78} = -\frac{3688}{78} = -\frac{1844}{39}$$ 2. Solve the equation: $$\frac{1}{11}(112 - 31x) = \frac{1}{7}(67 + 14x)$$ Step 1: Multiply both sides by 77 (LCM of 11 and 7): $$7(112 - 31x) = 11(67 + 14x)$$ $$784 - 217x = 737 + 154x$$ Step 2: Rearrange: $$784 - 737 = 154x + 217x$$ $$47 = 371x$$ $$x = \frac{47}{371}$$ 3. Simplify the expression: $$9(3x + 2y)^2 - 8(5x - 7y)^2$$ Step 1: Expand each square: $$(3x + 2y)^2 = 9x^2 + 12xy + 4y^2$$ $$(5x - 7y)^2 = 25x^2 - 70xy + 49y^2$$ Step 2: Multiply by coefficients: $$9(9x^2 + 12xy + 4y^2) = 81x^2 + 108xy + 36y^2$$ $$-8(25x^2 - 70xy + 49y^2) = -200x^2 + 560xy - 392y^2$$ Step 3: Combine: $$(81x^2 - 200x^2) + (108xy + 560xy) + (36y^2 - 392y^2) = -119x^2 + 668xy - 356y^2$$ 4. Factor the polynomial: $$2x^6 - 28x^5 - 64x^4$$ Step 1: Factor out the greatest common factor: $$2x^4(x^2 - 14x - 32)$$ Step 2: Factor the quadratic: $$x^2 - 14x - 32 = (x - 16)(x + 2)$$ Final factorization: $$2x^4(x - 16)(x + 2)$$