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

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3 — Easy The function is defined on transformed inputs. Substitute $a=3x+2y$, $b=2x-5y$, solve for $x$ in terms of $a,b$, and that rewrites $f$ explicitly. Then feed in $(x,2x)$ and set equal to $27$. 1 Invert the substitution. Let $a=3x+2y$ and $b=2x-5y$. Solving this $2	imes 2$ system: $$x=\frac{5a+2b}{19},\qquad y=\frac{2a-3b}{19}$$ 2 Rewrite $f$ explicitly. Since $f(a,b)=19x$: $$f(a,b)=19\cdot\frac{5a+2b}{19}=5a+2b$$ 3 Apply to $(x,2x)$ and set equal to $27$: $$\begin{aligned} &f(x,2x)=5x+2(2x)

CAT 2023 Slot 3 — QA Question 22

Answer & solution