1. **State the problem:** Solve the simultaneous equations: $$\frac{4}{7x - 4} = \frac{1}{6y}$$ and $$\frac{5x}{3y + 2} = 4$$ 2. **Rewrite the equations:** From the first equation: $$\frac{4}{7x - 4} = \frac{1}{6y}$$ Cross-multiply to get: $$4 \cdot 6y = 1 \cdot (7x - 4)$$ which simplifies to: $$24y = 7x - 4$$ From the second equation: $$\frac{5x}{3y + 2} = 4$$ Multiply both sides by $(3y + 2)$: $$5x = 4(3y + 2)$$ which expands to: $$5x = 12y + 8$$ 3. **Express $x$ from the second equation:** $$x = \frac{12y + 8}{5}$$ 4. **Substitute $x$ into the first equation:** $$24y = 7\left(\frac{12y + 8}{5}\right) - 4$$ Multiply both sides by 5 to clear the denominator: $$5 \cdot 24y = 7(12y + 8) - 20$$ which is: $$120y = 84y + 56 - 20$$ Simplify the right side: $$120y = 84y + 36$$ 5. **Solve for $y$:** Subtract $84y$ from both sides: $$120y - 84y = 36$$ $$36y = 36$$ Divide both sides by 36: $$y = 1$$ 6. **Find $x$ using $y=1$:** $$x = \frac{12(1) + 8}{5} = \frac{12 + 8}{5} = \frac{20}{5} = 4$$ **Final answer:** $$x = 4, \quad y = 1$$