Question

Y+1 y+8 y+2 y+7

Answer 1

$\dfrac{y+1}{y+2}+\dfrac{y+8}{y+9}=\dfrac{y+2}{y+3}+\dfrac{y+7}{y+8}$

Write all fractions in terms of a common denominator:

$\dfrac{y+1}{y+2}=\dfrac{(y+1)(y+9)(y+3)(y+8)}{(y+2)(y+9)(y+3)(y+8)}$

$\dfrac{y+8}{y+9}=\dfrac{(y+8)^2(y+2)(y+3)}{(y+2)(y+9)(y+3)(y+8)}$

$\dfrac{y+2}{y+3}=\dfrac{(y+2)^2(y+9)(y+8)}{(y+2)(y+9)(y+3)(y+8)}$

$\dfrac{y+7}{y+8}=\dfrac{(y+7)(y+2)(y+9)(y+3)}{(y+2)(y+9)(y+3)(y+8)}$

Then move all fractions to one side and simplify the numerator:

$\dfrac{(y+1)(y+9)(y+3)(y+8)+(y+8)^2(y+2)(y+3)-(y+2)^2(y+9)(y+8)-(y+7)(y+2)(y+9)(y+3)}{(y+2)(y+9)(y+3)(y+8)}=0$

The numerator dictates when the fraction reduces to 0. The denominator can never be 0, so we know that y cannot take any of the values -2, -9, -3, nor -8.

So the equation reduces to

$(y+1)(y+9)(y+3)(y+8)+(y+8)^2(y+2)(y+3)-(y+2)^2(y+9)(y+8)-(y+7)(y+2)(y+9)(y+3)=0$

Expand the left side; you would end up with

$-6(2y+11)=0$

$2y+11=0$

$2y=-11$

$\implies\boxed{y=-\dfrac{11}2}$