If $3^{(x - y)} = 27$ and $3^{(x + y)} = 243$, then $x$ is equal to

### Concept & Logic To solve exponential equations, express both sides of the equation with the exact same base. Once the bases match, you can directly equate their exponents to form a system of linear equations. ### Step-by-Step Solution * **Step 1:** Express the numbers on the right side of both equations as powers of the base $3$. We know that $27 = 3 \times 3 \times 3 = 3^3$. We know that $243 = 27 \times 9 = 3^3 \times 3^2 = 3^5$. * **Step 2:** Rewrite the original equations with matching bases. Equation 1: $3^{(x - y)} = 3^3$ Equation 2: $3^{(x + y)} = 3^5$ * **Step 3:** Equate the exponents, as the bases ($3$) are equal. (1) $x - y = 3$ (2) $x + y = 5$ * **Step 4:** Solve the resulting system of linear equations. The easiest method here is addition. Add equation (1) and equation (2) together: $(x - y) + (x + y) = 3 + 5$ $2x = 8$ * **Step 5:** Isolate $x$. $x = \frac{8}{2}$ $x = 4$ ### Exam Strategy & Shortcut **Mental Math Elimination:** The powers of 3 are standard knowledge ($3, 9, 27, 81, 243$). You can instantly read the equations as "sum is 5, difference is 3". What two numbers add to 5 and subtract to 3? They must be 4 and 1. Since the question asks for $x$ (the larger number, since $x-y$ is positive), $x$ must be 4. This bypasses formal written steps entirely. ### Common Pitfall A common error is solving for the wrong variable in the rush of the exam. After setting up the equations, a student might subtract them to find $2y = 2 \rightarrow y = 1$, and then accidentally select an option matching $y$ if one were provided. Always double-check which variable the question is asking for. ### Final Answer Therefore, the correct answer is **4**.