Question

Find the value of the variable y, for which:

the sum of the fractions y+1/y−5 and 10/y+5 is equal to their product.

Answer 1

Answer: y = -11

Explanation:

`\text{Sum:}\\\\.\quad (y+1)/(y-5)+(10)/(y+5)\\\\\\=\bigg((y+5)/(y+5)\bigg)(y+1)/(y-5)+(10)/(y+5)\bigg((y-5)/(y-5)\bigg)\\\\\\=(y^2+6y+5)/((y+5)(y-5))+(10y-50)/((y+5)(y-5))\\\\\\=(y^2+6y+5+10y-50)/((y+5)(y-5))\\\\\\=(y^2+16y-45)/((y+5)(y-5))`

`\text{Product:}\\\\.\quad ((y+1)(10))/((y-5)(y+5))\\\\\\=(10y+10)/((y-5)(y+5))\\\\\\\text{Sum = Product:}\\\\(y^2+16y-45)/((y+5)(y-5))=(10y+10)/((y+5)(y-5))\qquad Restriction: y\\eq -5, 5\\\\\\\rightarrow y^2+16y-45=10y+10\\\\\rightarrow y^2+6y-55=0\\\\\rightarrow (y+11)(y-5)=0\\\\\rightarrow y+11=0\quad and\quad y-5=0\\\\\rightarrow y=-11\qquad and\qquad y=5\\\\\\\text{Since y = 5 is a restricted value, it is not a valid solution}`