0\\\\t+\frac{1}{t}=3,\; t^2-3t+1=0,\\\\D=9-4=5,\; t_1=\frac{3-\sqrt5}{2},\; t_2=\frac{3+\sqrt5}{2}\\\\\sqrt{\frac{x}{y}}=\frac{3-\sqrt5}{2},\; \frac{x}{y}=(\frac{3-\sqrt5}{2})^2=\frac{14-6\sqrt5}{4}\\\\x+y=41\; \to \; y=41-x\; \to \; \frac{x}{41-x}=\frac{14-6\sqrt5}{4}\\\\4x=(41-x)(14-6\sqrt5)\\\\4x=574-246\sqrrt5-14x+6\sqrt5x" alt=" \left \{ {{x+y=41} \atop {\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}=3}} \right.\\\\t=\sqrt{\frac{x}{y}},\; \frac{1}{t}=\sqrt{\frac{y}{x}},\; x\ne 0,\; y\ne 0,\; t>0\\\\t+\frac{1}{t}=3,\; t^2-3t+1=0,\\\\D=9-4=5,\; t_1=\frac{3-\sqrt5}{2},\; t_2=\frac{3+\sqrt5}{2}\\\\\sqrt{\frac{x}{y}}=\frac{3-\sqrt5}{2},\; \frac{x}{y}=(\frac{3-\sqrt5}{2})^2=\frac{14-6\sqrt5}{4}\\\\x+y=41\; \to \; y=41-x\; \to \; \frac{x}{41-x}=\frac{14-6\sqrt5}{4}\\\\4x=(41-x)(14-6\sqrt5)\\\\4x=574-246\sqrrt5-14x+6\sqrt5x" align="absmiddle" class="latex-formula">
