Question

6．解方程组：$\left\{\begin{array}{l}{{x}^{2}-9{y}^{2}=0}\\{{x}^{2}-2xy+{y}^{2}=6}\end{array}\right.$．

Answer 1

分析 首先对原方程组进行化简，然后分别重新组合，成为4个方程组，最后解这两个方程组即可．

解答 解：原方程组可化为如下4个方程组
（1）$\left\{\begin{array}{l}{x+3y=0}\\{x-y=-\sqrt{6}}\end{array}\right.$，（2）$\left\{\begin{array}{l}{x+3y=0}\\{x-y=\sqrt{6}}\end{array}\right.$，（3）$\left\{\begin{array}{l}{x-3y=0}\\{x-y=-\sqrt{6}}\end{array}\right.$，（4）$\left\{\begin{array}{l}{x-3y=0}\\{x-y=\sqrt{6}}\end{array}\right.$，
解方程组（1）得$\left\{\begin{array}{l}{x=-\frac{3\sqrt{6}}{4}}\\{y=\frac{\sqrt{6}}{4}}\end{array}\right.$，
解方程组（2）得$\left\{\begin{array}{l}{x=\frac{3\sqrt{6}}{4}}\\{y=-\frac{\sqrt{6}}{4}}\end{array}\right.$，
解方程组（3）得$\left\{\begin{array}{l}{x=-\frac{3\sqrt{6}}{2}}\\{y=-\frac{\sqrt{6}}{2}}\end{array}\right.$，
解方程组（4）得$\left\{\begin{array}{l}{x=\frac{3\sqrt{6}}{2}}\\{y=\frac{\sqrt{6}}{2}}\end{array}\right.$．
故原方程组的解为$\left\{\begin{array}{l}{x=-\frac{3\sqrt{6}}{4}}\\{y=\frac{\sqrt{6}}{4}}\end{array}\right.$，$\left\{\begin{array}{l}{x=\frac{3\sqrt{6}}{4}}\\{y=-\frac{\sqrt{6}}{4}}\end{array}\right.$，$\left\{\begin{array}{l}{x=-\frac{3\sqrt{6}}{2}}\\{y=-\frac{\sqrt{6}}{2}}\end{array}\right.$，$\left\{\begin{array}{l}{x=\frac{3\sqrt{6}}{2}}\\{y=\frac{\sqrt{6}}{2}}\end{array}\right.$．

点评 本题主要考查解高次方程，关键在于对原方程组的两个方程进行化简，重新组合．