Discussion

Explanation:

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

Let
3x + 2y = a   ...(1) and
2x - 5y = b   ...(2)

Solving (1) and (2), we get
x = (5a + 2b)/19 and y = (2a - 3b)/19

∴ f(3x + 2y, 2x - 5y) = 19x can be rewritten as

f(a, b) = 5a + 2b

Now, substituting a = x and b = 2x, we get
​​​​​​​⇒ f(x, 2x) = 5x + 4x = 9x
​​​​​​​⇒ 27 = 5x + 4x = 9x
​​​​​​​⇒ x = 3

Hence, 3.

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