Question

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

Accepted Answer

3. We are given $f(3x+2y, 2x-5y) = 19x$. To find $f(X, 2X)$, set: 1) $3x + 2y = X$ 2) $2x - 5y = 2X$ Multiplying the first by 5 and the second by 2: $$15x + 10y = 5X$$ $$4x - 10y = 4X$$ Adding these equations gives: $$19x = 9X$$ Thus, $f(X, 2X) = 19x = 9X$. Given $f(X, 2X) = 27$: $$9X = 27 \implies X = 3$$ So the required value of $x$ is $3$.

Functional Equation with Two Variables

Explanation

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