Question

已知y/x=3/4,求x/（x+y）+y/（x-y）-y²/（x²-y²）的值。

如题。已知y/x=3/4,求x/（x+y）+y/（x-y）-y²/（x²-y²）的值。
为什么自己算出来是89/48？

Answer 1

∵y/x=3/4
设比的每一份为t
x=4t,y=3t
x/（x+y）+y/（x-y）-y²/（x²-y²）
=(x²-xy+xy+y²)/[(x+y)(x-y)]-y²/（x²-y²）
=(x²+y²)/（x²-y²)-y²/（x²-y²）
=x²/(x²-y²)=16t²/(16t²-9t²)=16/7

Answer 2

y/x=3/4
所以 x/y=4/3 x²/y²=16/9

x/（x+y）+y/（x-y）-y²/（x²-y²）
通分
=x(x-y)/(x²-y²)+y(x+y)/(x²-y²)-y²/(x²-y²)
=[x(x-y)+y(x+y)-y²]/(x²-y²)
=(x²-xy+xy+y²-y²)/(x²-y²)
=x²/(x²-y²)
上下同时除以y²
=(x²/y²)/(x²/y²-1)
=(16/9)/(16/9-1)
=(16/9)/(7/9)
=16/7

Answer 3

x/（x+y）+y/（x-y）-y²/（x²-y²）=( x²-xy)/（x²-y²) + (xy+y²) /（x²-y²)-y²/（x²-y²) =x² /（x²-y²)

(x²-y²) /x²= 1-y²/x²=1-9/16 = 7/16, 所以x² /（x²-y²)=16/7