1. **State the problem:** We are given the system of equations: $$3^x + y = \sqrt[3]{27}$$ $$4^y \div 2^x = 8$$ We need to find the values of $x$ and $y$. 2. **Simplify the constants:** $$\sqrt[3]{27} = 3$$ because $3^3 = 27$. So the first equation becomes: $$3^x + y = 3$$ 3. **Rewrite the second equation:** Recall that $4 = 2^2$, so: $$4^y = (2^2)^y = 2^{2y}$$ The second equation is: $$\frac{4^y}{2^x} = 8$$ Substitute: $$\frac{2^{2y}}{2^x} = 8$$ Using the law of exponents: $$2^{2y - x} = 8$$ Since $8 = 2^3$, we have: $$2^{2y - x} = 2^3$$ Therefore: $$2y - x = 3$$ 4. **Rewrite the first equation:** $$3^x + y = 3$$ We want to express $y$ in terms of $x$: $$y = 3 - 3^x$$ 5. **Substitute $y$ into the second equation:** $$2(3 - 3^x) - x = 3$$ Simplify: $$6 - 2 \cdot 3^x - x = 3$$ Bring all terms to one side: $$6 - 2 \cdot 3^x - x - 3 = 0$$ $$3 - 2 \cdot 3^x - x = 0$$ 6. **Solve for $x$:** Try integer values for $x$: - For $x=0$: $$3 - 2 \cdot 3^0 - 0 = 3 - 2 - 0 = 1 \neq 0$$ - For $x=1$: $$3 - 2 \cdot 3^1 - 1 = 3 - 6 - 1 = -4 \neq 0$$ - For $x=2$: $$3 - 2 \cdot 3^2 - 2 = 3 - 18 - 2 = -17 \neq 0$$ - For $x=-1$: $$3 - 2 \cdot 3^{-1} - (-1) = 3 - 2 \cdot \frac{1}{3} + 1 = 3 - \frac{2}{3} + 1 = \frac{10}{3} \neq 0$$ - For $x=3$: $$3 - 2 \cdot 3^3 - 3 = 3 - 54 - 3 = -54 \neq 0$$ Try $x=1.5$: $$3 - 2 \cdot 3^{1.5} - 1.5$$ Calculate $3^{1.5} = 3^{1 + 0.5} = 3 \times \sqrt{3} \approx 3 \times 1.732 = 5.196$ So: $$3 - 2 \times 5.196 - 1.5 = 3 - 10.392 - 1.5 = -8.892 \neq 0$$ Try $x=0.5$: $$3 - 2 \times 3^{0.5} - 0.5 = 3 - 2 \times 1.732 - 0.5 = 3 - 3.464 - 0.5 = -0.964 \neq 0$$ Try $x=0.3$: $$3 - 2 \times 3^{0.3} - 0.3$$ Calculate $3^{0.3} = e^{0.3 \ln 3} \approx e^{0.329} = 1.39$ So: $$3 - 2 \times 1.39 - 0.3 = 3 - 2.78 - 0.3 = -0.08 \approx 0$$ Try $x=0.25$: $$3 - 2 \times 3^{0.25} - 0.25$$ Calculate $3^{0.25} = e^{0.25 \ln 3} \approx e^{0.274} = 1.315$ So: $$3 - 2 \times 1.315 - 0.25 = 3 - 2.63 - 0.25 = 0.12 \neq 0$$ By interpolation, $x \approx 0.28$. 7. **Find $y$:** $$y = 3 - 3^x \approx 3 - 3^{0.28}$$ Calculate $3^{0.28} = e^{0.28 \ln 3} \approx e^{0.307} = 1.36$ So: $$y \approx 3 - 1.36 = 1.64$$ **Final answer:** $$x \approx 0.28, \quad y \approx 1.64$$