Suppose f(x,y) is a real-valued function such that f(3x+2y,2x-5y)=19x, for all real numbers x and y. The value of x for which f(x,2x)=27, is:

Question

UPSC Quizroom · Accepted Answer

Let the original variables in the definition be p and q. We want f(x, 2x) = 27.

Step 1: Solve for p and q such that:

3p + 2q = x
2p − 5q = 2x

Step 2: Solve for q from the first equation:

2q = x − 3p → q = (x − 3p)/2

Step 3: Substitute into the second equation:

2p − 5*((x − 3p)/2) = 2x
Multiply both sides by 2: 4p − 5x + 15p = 4x → 19p − 5x = 4x → 19p = 9x → p = 9x / 19

Step 4: Use definition of f:

f(3p + 2q, 2p − 5q) = 19p = 19*(9x / 19) = 9x

We are given f(x, 2x) = 27 → 9x = 27 → x = 3

Answer: 3

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