1. **State the problem:** Solve the simultaneous equations: $$2^x = 4^{7 - 2y}$$ $$3^{5x - 2y} = 81$$ 2. **Rewrite the equations with common bases:** Note that $4 = 2^2$ and $81 = 3^4$. So rewrite the equations as: $$2^x = (2^2)^{7 - 2y} = 2^{2(7 - 2y)} = 2^{14 - 4y}$$ $$3^{5x - 2y} = 3^4$$ 3. **Set the exponents equal since bases are the same:** From the first equation: $$x = 14 - 4y$$ From the second equation: $$5x - 2y = 4$$ 4. **Substitute $x$ from the first into the second:** $$5(14 - 4y) - 2y = 4$$ Simplify: $$70 - 20y - 2y = 4$$ $$70 - 22y = 4$$ 5. **Isolate $y$:** $$-22y = 4 - 70$$ $$-22y = -66$$ $$y = \frac{\cancel{-66}}{\cancel{-22}} = 3$$ 6. **Find $x$ using $y=3$:** $$x = 14 - 4(3) = 14 - 12 = 2$$ **Final answer:** $$x = 2, \quad y = 3$$