If $$3^{x-y}=27$$ and $$3^{x+y}=243$$, then the value of $$x$$ is

Appeared in: UPPSC-2021

Explanation

The problem involves two exponential equations with the same base (3).
By expressing 27 as 3³ and 243 as 3⁵, we can equate the exponents to form a system of two linear equations: x - y = 3 and x + y = 5.
Adding these two equations eliminates the 'y' variable: (x - y) + (x + y) = 3 + 5, which simplifies to 2x = 8.
Solving for x gives x = 4.

Option A: This value does not satisfy the derived equations. If x=0, then 0-y=3 (y=-3) and 0+y=5 (y=5), which is a contradiction.
Option B: If x=2, then from the first equation, 2-y=3, which means y=-1. From the second equation, 2+y=5, which means y=3. Since y cannot be both -1 and 3, this option is incorrect.
Option D: This is the value of 2x, not x. After adding the two equations (x-y=3 and x+y=5) to get 2x=8, a final step of dividing by 2 is required to find x.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Solving a system of linear equations derived from exponential equations as background academic context rather than a clinical decision trigger.
This is a general mathematics question testing algebraic skills, not a clinical concept.
Proficiency in basic mathematics is essential for various nursing calculations, such as drug dosage, IV drip rates, and body mass index (BMI).
What if the second equation was 3^(x+y) = 81? Then x+y = 4. Adding this to x-y=3 would give 2x=7, and x=3.5. This shows how changing one value alters the entire solution.

How to Approach the Question

First, identify that both equations share a common base (3).
The next step is to rewrite the numbers on the right side of the equations (27 and 243) as powers of this common base, 3.
Apply the fundamental rule of exponents: if a^m = a^n, then the exponents must be equal (m = n). This will transform the two exponential equations into two simple linear equations.
You will now have a system of two linear equations with two variables (x and y).
Use the elimination method by adding the two equations together. This will cancel out the 'y' variable, leaving an equation solely in terms of 'x'.
Solve the resulting equation for 'x' and select the corresponding option.

Concept Tested & Keywords

Concept Tested: Solving a system of linear equations derived from exponential equations.
Stem keywords: 3^{x-y}=27, 3^{x+y}=243, value of x
Lead-in keywords: BEST, MOST RELEVANT CLUE
Negative lead-in flag: false

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