Question

解方程：(x+1)/(x+2)+(x+6)/(x+7)=(x+2)/(x+3)+(x+5)/(x+6)

Answer 1

原方程可化为：
1 - 1/(x+2) + 1 - 1/(x+7)=1 - 1/(x+3) + 1 - 1/(x+6)
即有：1/(x+3) - 1/(x+2)=1/(x+7) - 1/(x+6)
左右两边分别通分得：-1/[(x+3)(x+2)]=-1/[(x+7)(x+6)]
则有：(x+3)(x+2)=(x+7)(x+6)
x²+5x+6=x²+13x+42
移项整理得：8x=-36
解得：x=-4.5
经检验，原方程的解为：x=-4.5

Answer 2

(x+1)/(x+2)+(x+6)/(x+7)=(x+2)/(x+3)+(x+5)/(x+6)
∴（x+2-1）/（x+2）+(x+7-1)/(x+7) =(x+3-1)/(x+3) +(x+6-1)/(x+6)
∴1-1/(x+2)+1-1/(x+7)=1-1/(x+3)+1-1/(x+6)
∴-1/(x+2)-1/(x+7)=-1/(x+3)-1/(x+6)
∴1/(x+2)+1/(x+7)=1/(x+3)+1/(x+6)
∴1/(x+2)-1/(x+3)=1/(x+6)-1/(x+7)
∴(x+3-(x+2))/(x+2)(x+3)=(x+7-(x+6))/(x+6)(x+7)
∴1/(x+2)(x+3)=1/(x+6)(x+7)
∴(x+2)(x+3)=(x+6)(x+7)
∴x²+5x+6=x²+13x+42
∴8x=-36
∴x=-9/2