Question

20．已知方程组$\left\{\begin{array}{l}{5x+y=7}\\{ax+5y=3}\end{array}\right.$与方程组$\left\{\begin{array}{l}{x-2y=8}\\{5x+by=2}\end{array}\right.$有相同的解，求a，b的值．

Answer 1

分析 解方程组$\left\{\begin{array}{l}{5x+y=7}\\{x-2y=8}\end{array}\right.$求得x，y的值，然后把x，y的值代入ax+5y=3和5x+by=2即可得到一个关于a，b的方程组，解方程组即可求得a，b的值．

解答 解：因为方程组$\left\{\begin{array}{l}{5x+y=7}\\{ax+5y=3}\end{array}\right.$与方程组$\left\{\begin{array}{l}{x-2y=8}\\{5x+by=2}\end{array}\right.$有相同的解，
所以它们的解是方程组$\left\{\begin{array}{l}{5x+y=7}\\{x-2y=8}\end{array}\right.$的解，
解方程组$\left\{\begin{array}{l}{5x+y=7}\\{x-2y=8}\end{array}\right.$得$\left\{\begin{array}{l}{x=2}\\{y=-3}\end{array}\right.$，
将$\left\{\begin{array}{l}{x=2}\\{y=-3}\end{array}\right.$代入$\left\{\begin{array}{l}{ax+5y=3}\\{5x+by=2}\end{array}\right.$得$\left\{\begin{array}{l}{2a-15=3}\\{10-3b=2}\end{array}\right.$，解得$\left\{\begin{array}{l}{a=9}\\{b=\frac{8}{3}}\end{array}\right.$．

点评 本题考查了方程组的解的定义，正确根据定义转化成解方程组的问题是关键．