Given $2^x = 8^{y+1}$ and $9^y = 3^{x-9}$, the value of $x+y$ is

### Concept & Formula This problem requires converting separate exponential equations into a system of linear equations by matching their prime bases. Rule: If $a^m = a^n$, then $m = n$ (given $a > 0$ and $a \neq 1$). ### Step-by-Step Solution * **Step 1:** Simplify the first equation by writing both sides with a base of $2$. Given: $ 2^x = 8^{y+1} $ Since $ 8 = 2^3 $: $ 2^x = (2^3)^{y+1} $ $ 2^x = 2^{3y+3} $ Equating the exponents gives our first linear equation: (1) $ x = 3y + 3 $ * **Step 2:** Simplify the second equation by writing both sides with a base of $3$. Given: $ 9^y = 3^{x-9} $ Since $ 9 = 3^2 $: $ (3^2)^y = 3^{x-9} $ $ 3^{2y} = 3^{x-9} $ Equating the exponents gives our second linear equation: (2) $ 2y = x - 9 $ * **Step 3:** Solve the linear system using substitution. Substitute equation (1) into equation (2): $ 2y = (3y + 3) - 9 $ $ 2y = 3y - 6 $ * **Step 4:** Isolate $y$. $ 6 = 3y - 2y $ $ y = 6 $ * **Step 5:** Substitute $y=6$ back into equation (1) to find $x$. $ x = 3(6) + 3 $ $ x = 18 + 3 = 21 $ * **Step 6:** Calculate the final required value, $x+y$. $ x + y = 21 + 6 = 27 $ ### Exam Strategy & Shortcut **Reverse Engineering from Options:** If you derive $x = 3y+3$ quickly, notice that $x$ is approximately 3 times $y$. So $x+y$ is roughly $4y$. Since $4y \approx \text{Total}$, test $y$-values that yield integer $x$ values. Try $y=5 \implies x=18 \implies \text{Sum}=23$ (not an option). Try $y=6 \implies x=21 \implies \text{Sum}=27$ (Option d). Verify with the second equation: $2(6) = 21 - 9 \implies 12 = 12$. This confirms 27 is the correct answer rapidly without full algebraic substitution. ### Common Pitfall The most frequent error is calculating the values of $x$ (21) or $y$ (6) and instinctively looking for those numbers in the options, forgetting that the question specifically asks for the sum $x+y$. Always re-read the final sentence of the prompt before selecting an answer. ### Final Answer Therefore, the correct answer is **27**.