Question

cho hpt sau

\(\left\{{}\begin{matrix}x+\sqrt{y-1}=1\\y+\sqrt{x-1}=1\end{matrix}\right.\) có bao nhiêu cặp nghiệm (x,y)

Answer 1

\(\left\{{}\begin{matrix}x+\sqrt{y-1}=1\\y+\sqrt{x-1}=1\end{matrix}\right.\left(1\right)\left(ĐK:x-1\ge0;y-1\ge0\right)\\ \Leftrightarrow\left\{{}\begin{matrix}x-1+\sqrt{y-1}=0\\y-1+\sqrt{x-1}=0\end{matrix}\right.\)

Do x-1\(\ge\)0, y-1\(\ge\)0 nên :

\(\left\{{}\begin{matrix}x-1+\sqrt{y-1}=0\\y-1+\sqrt{x-1}=0\end{matrix}\right.=0\Leftrightarrow\left\{{}\begin{matrix}x-1=0\\y-1=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=1\end{matrix}\right.\) (Thỏa ĐK)

Vậy HPT có 1 cặp nghiệm (x;y).