Question

x ^ 2 + y ^ 2 = 25 x + y + xy = 19 求x,y所有的解，要解答过程。谢谢！

RT

Answer 1

x ^ 2 + y ^ 2 = （x+y)^2 - 2xy = 25
设 x+y = m ,xy = n
则 化为 m^2 - 2n = 25
m + n =19, n = 19 - m,代入上式
m^2 - 2(19 - m) = 25
m^2 + 2m - 63 = 0
(m-7)(m+9)=0
m = 7,n = 12 或 m = -9,n=28
所以两种情况：
1. x+y= 7, xy = 12
解得 x=3,y=4 或 x=4,y=3
2.x+y= -9, xy = 28
x^2 + 9x + 28 =0,无解

综上， x=3,y=4 或 x=4,y=3

Answer 2

x + y + xy = 19
x + y = 19 - xy
两边平方得：
x² + y² + 2xy = x²y² - 38xy + 361
25 + 2xy = x²y² - 38xy + 361
x²y² - 40xy + 336 = 0
(xy - 12)(xy + 28) = 0
xy = 12 或 xy = -28

当 xy = 12时， x + y = 7
所以 x = 3 , y = 4 或 x = 4 , y = 3

当 xy = -28时， x + y = 47
x = (47 + √2321)/2 , y = (47 - √2321)/2
或
x = (47 + √2321)/2 , y = (47 - √2321)/2

Answer 3

(x+y)^2=(19-xy)^2
(xy)^2-40xy+336=0
xy=12 或 xy=28
当xy=12时
x+y=7
x=3 y=4 或x=4 y=3
当xy=28时
x+y=-9
无实数解