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Functional Equation with Two Variables

CAT 2023 Slot 3 · Quantitative Ability · Medium · Algebra

This is a medium Quantitative Ability question from the CAT 2023 Slot 3 paper. It tests Algebra. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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

Answer

3

Explanation

We are given f(3x+2y,2x−5y)=19xf(3x+2y, 2x-5y) = 19x.

To find f(X,2X)f(X, 2X), set:

  1. 3x+2y=X3x + 2y = X
  2. 2x−5y=2X2x - 5y = 2X

Multiplying the first by 5 and the second by 2: 15x+10y=5X15x + 10y = 5X 4x−10y=4X4x - 10y = 4X Adding these equations gives: 19x=9X19x = 9X

Thus, f(X,2X)=19x=9Xf(X, 2X) = 19x = 9X.

Given f(X,2X)=27f(X, 2X) = 27: 9X=27  ⟹  X=39X = 27 \implies X = 3

So the required value of xx is 33.

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