Question

If \(3^{(x-y)} = 27\) and \(3^{(x+y)} = 243\), find the value of \(x\).

• A. \(4\)
• B. \(6\)
• C. \(2\)
• D. \(0\)

Hint:

When given equations with the same base in exponent form, equate exponents directly after rewriting the numbers as powers of that base.

Answer 1

The Correct Option is A

Solution:
Step 1 (Rewrite the powers in terms of 3).
Given: \[ 3^{(x-y)} = 27 \quad\text{and}\quad 3^{(x+y)} = 243 \] We know \(27 = 3^3\) and \(243 = 3^5\). Thus: \[ 3^{(x-y)} = 3^3 \quad\quad x - y = 3 \] \[ 3^{(x+y)} = 3^5 \quad\quad x + y = 5 \] Step 2 (Solve the system of equations).
From: \[ x - y = 3 \quad\text{(1)} \] \[ x + y = 5 \quad\text{(2)} \] Add (1) and (2): \[ 2x = 8 \quad\quad x = 4 \] \[ {4 \ \text{(Option (a)}} \]