Question

17．已知x，y为整数并且满足等式x²+y²+1=2x+2y，求x+y的值．

Answer 1

分析 首先根据x²+y²+1=2x+2y，推得（x-1）²+（y-1）²=1；然后根据x、y都为整数，分类讨论，分别求出x、y的值，再把它们求和，求出x+y的值是多少即可．

解答 解：因为x²+y²+1=2x+2y，
所以（x-1）²+（y-1）²=1，
因为x，y为整数，
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）$\left\{\begin{array}{l}{x=2}\\{y=1}\end{array}\right.$时，
x+y=2+1=3；

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

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

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

点评 此题主要考查了用字母表示数的方法，要熟练掌握，解答此题的关键是判断出：（x-1）²+（y-1）²=1．