Question

What is the solution of -8/2y-8=5/y+4 - 7y+8/y^2-16

Answer 1

y = 8

Explanation:

`Domain:\\\\2y-8\\eq0\ \wedge\ y+4\\eq0\ \wedge\ y^2-16\\eq0\\\\2y\\eq8\ \wedge\ y\\eq-4\ \wedge\ y^2\\eq16\\\\y\\eq4\ \wedge\ y\\eq-4\ \wedge\ y\\eq\pm√(16)\\\\y\\eq4\ \wedge\ y\\eq-4\ \wedge\ y\\eq-4\ \wedge\ y\\eq4\\\\\boxed{y\\eq-4\ \wedge\ y\\eq4}\\\\===========================`

`(-8)/(2y-8)=(5)/(y+4)-(7y+8)/(y^2-16)\\\\(-8)/(2(y-4))=(5)/(y+4)-(7y+8)/(y^2-4^2)\qquad\text{use}\ a^2-b^2=(a-b)(a+b)\\\\(-8)/(2(y-4))=(5)/(y+4)-(7y+8)/((y-4)(y+4))\qquad\text{multiply both sides by (-2)}\\\\(8)/(y-4)=-(10)/(y+4)+(14y+16)/((y-4)(y+4))\qquad\text{add}\ (10)/(y+4)\ \text{to both sides}\\\\(8)/(y-4)+(10)/(y+4)=(14y+16)/((y-4)(y+4))`

`(8(y+4))/((y-4)(y+4))+(10(y-4))/((y-4)(y+4))=(14y+16)/((y-4)(y+4))\\\\(8(y+4)+10(y-4))/((y-4)(y+4))=(14y+16)/((y-4)(y+4))\qquad\text{use the distributive property}\\\\(8y+32+10y-40)/((y-4)(y+4))=(14y+16)/((y-4)(y+4))\qquad\text{combine like terms}\\\\((8y+10y)+(32-40))/((y-4)(y+4))=(14y+16)/((y-4)(y+4))\\\\(18y-8)/((y-4)(y+4))=(14y+16)/((y-4)(y+4))\iff18y-8=14y+16\\\\18y-8=14y+16\qquad\text{subtract 14y from both sides}`

`4y-8=16\qquad\text{add 8 to both sides}\\\\4y=24\qquad\text{divide both sides by 4}\\\\y=8\in D`

Answer 2

6

Explanation: