Final answer:
To solve the given equation, we simplify it by multiplying both sides by (y+4)(y-4), cancel out common terms, distribute, combine like terms, move all terms to one side, factor the quadratic equation, and solve for y. The solutions are y = -2.25 and y = 6. None of the given options A, B, C, or D are true.
Step-by-step explanation:
To solve the given equation, -8/2y-8=5/y+4-7y+8/y²-16, we need to simplify the equation first. Multiplying both sides of the equation by (y+4)(y-4) will help us eliminate the denominator.
Multiply the entire equation by (y+4)(y-4): (y+4)(y-4)(-8/2y-8) = (y+4)(y-4)(5/y+4-7y+8/y²-16)
Simplify both sides of the equation:
On the left side, we can cancel out (y+4) in the denominator: -8(y-4) = 5(y+4-7y+8)/(y-4)(y+4)
On the right side, we can simplify the expression: (y+4-7y+8) = 5(y+4)/(y-4)
Distribute and simplify further:
On the left side, distribute -8 to (y-4): -8y + 32 = (y+4)/(y-4)
On the right side, distribute 5 to (y+4): (5y + 20)/(y-4)
Combine like terms: -8y + 32 = (5y + 20)/(y-4)
Multiply both sides by (y-4) to eliminate the fraction: (y-4)(-8y + 32) = (5y + 20)
Expand and simplify: -8y^2 + 32y - 4(-8y) + 4(32) = 5y + 20
Combine like terms: -8y^2 + 32y + 32y + 128 = 5y + 20
Combine like terms: -8y^2 + 64y + 128 = 5y + 20
Moving all terms to one side: -8y^2 + 64y + 128 - 5y - 20 = 0
Combine like terms: -8y^2 + 59y + 108 = 0
Factor the quadratic equation: (4y + 9)(-2y + 12) = 0
Set each factor equal to zero: 4y + 9 = 0 or -2y + 12 = 0
Solve for y:
For 4y + 9 = 0: 4y = -9, y = -9/4, y = -2.25
For -2y + 12 = 0: -2y = -12, y = -12/-2, y = 6
Therefore, the solutions for the given equation are y = -2.25 and y = 6. None of the options A, B, C, or D are true.