Question

\(\left\{{}\begin{matrix}\sqrt{2x+y+1}-\sqrt{x+y}=1\\3x+2y=4\end{matrix}\right.\)

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Answer 1

\(PT\left(2\right)\Leftrightarrow y=\dfrac{4-3x}{2}\\ PT\left(1\right)\Leftrightarrow\sqrt{2x+\dfrac{4-3x}{2}+1}-\sqrt{x+\dfrac{4-3x}{2}}=1\\ \Leftrightarrow\sqrt{\dfrac{x+6}{2}}-\sqrt{\dfrac{4-x}{2}}=1\\ \Leftrightarrow\dfrac{\sqrt{x+6}-\sqrt{4-x}}{\sqrt{2}}=1\\ \Leftrightarrow\sqrt{x+6}=\sqrt{4-x}+\sqrt{2}\\ \Leftrightarrow x+6=6-x+2\sqrt{2\left(4-x\right)}\left(x\ge-6\right)\\ \Leftrightarrow x=\sqrt{8-2x}\\ \Leftrightarrow x^2=8-2x\left(x\ge0\right)\\ \Leftrightarrow x^2+2x-8=0\\ \Leftrightarrow x=2\left(x\ge0\right)\\ \Leftrightarrow y=\dfrac{4-3.2}{2}=-1\)

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