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

We need to find a pattern in the given functional equation to determine what the function $f(x,y)$ actually represents.

We're told that $f(3x + 2y, 2x - 5y) = 19x$ for all real numbers $x$ and $y$.

The key insight is to figure out what $f(u,v)$ equals in terms of $u$ and $v$.

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Let's work backwards from the expression $19x$. We need to see if $19x$ can be written as a linear combination of the two arguments $(3x + 2y)$ and $(2x - 5y)$.

Let's try: $a(3x + 2y) + b(2x - 5y) = 19x$

Expanding: $3ax + 2ay + 2bx - 5by = 19x$

Collecting terms: $(3a + 2b)x + (2a - 5b)y = 19x + 0y$

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For this to hold for all $x$ and $y$, we need:

Coefficient of $x$: $3a + 2b = 19$

Coefficient of $y$: $2a - 5b = 0$

From the second equation: $2a = 5b$, so $a = \tfrac{5b}{2}$

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Substituting into the first equation:

$3 \cdot \tfrac{5b}{2} + 2b = 19$

$\tfrac{15b}{2} + 2b = 19$

$\tfrac{15b + 4b}{2} = 19$

$\tfrac{19b}{2} = 19$

$b = 2$

Therefore: $a = \tfrac{5 \cdot 2}{2} = 5$

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This means $f(u,v) = 5u + 2v$

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Now we need to find $x$ such that $f(x, 2x) = 27$.

Using our discovered pattern:

$f(x, 2x) = 5x + 2(2x) = 5x + 4x = 9x$

Setting this equal to 27:

$9x = 27$

$x = 3$

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Answer: $x = 3$

Key Learning Point: When dealing with functional equations like this, the strategy is to express the given output as a linear combination of the inputs. This reveals the underlying structure of the function, making it easy to evaluate for any new inputs.