Question

17．已知x，y為整數(shù)并且滿足等式x²+y²+1=2x+2y，求x+y的值．

Answer 1

分析 首先根據(jù)x²+y²+1=2x+2y，推得（x-1）²+（y-1）²=1；然后根據(jù)x、y都為整數(shù)，分類討論，分別求出x、y的值，再把它們求和，求出x+y的值是多少即可．

解答 解：因?yàn)閤²+y²+1=2x+2y，
所以（x-1）²+（y-1）²=1，
因?yàn)閤，y為整數(shù)，
所以$\left\{\begin{array}{l}{x-1=1}\\{y-1=0}\end{array}\right.$、$\left\{\begin{array}{l}{x-1=-1}\\{y-1=0}\end{array}\right.$、$\left\{\begin{array}{l}{x-1=0}\\{y-1=1}\end{array}\right.$或$\left\{\begin{array}{l}{x-1=0}\\{y-1=-1}\end{array}\right.$，
解得$\left\{\begin{array}{l}{x=2}\\{y=1}\end{array}\right.$、$\left\{\begin{array}{l}{x=0}\\{y=1}\end{array}\right.$、$\left\{\begin{array}{l}{x=1}\\{y=2}\end{array}\right.$或$\left\{\begin{array}{l}{x=1}\\{y=0}\end{array}\right.$，
（1）$\left\{\begin{array}{l}{x=2}\\{y=1}\end{array}\right.$時(shí)，
x+y=2+1=3；

（2）$\left\{\begin{array}{l}{x=0}\\{y=1}\end{array}\right.$時(shí)，
x+y=0+1=1；

（3）$\left\{\begin{array}{l}{x=1}\\{y=2}\end{array}\right.$時(shí)，
x+y=1+2=3；

（4）$\left\{\begin{array}{l}{x=1}\\{y=0}\end{array}\right.$時(shí)，
x+y=1+0=1．
所以x+y的值是1或3．

點(diǎn)評(píng) 此題主要考查了用字母表示數(shù)的方法，要熟練掌握，解答此題的關(guān)鍵是判斷出：（x-1）²+（y-1）²=1．