What is the solution of StartFraction negative 8 Over 2 y minus 8 EndFraction = StartFraction 5 Over y + 4 EndFraction minus StartFraction 7 y + 8 Over y squared minus 16 EndFraction?
To solve for y, we can start by simplifying both sides of the equation:
Starting with the left side:
StartFraction -8 Over 2y - 8 EndFraction = StartFraction -8(1/y - 1) EndFraction = StartFraction -8(1/y - y/y) EndFraction = StartFraction -8(1 - y)/y EndFraction = StartFraction -8(1 - y) Over y EndFraction = StartFraction -8 + 8y Over y EndFraction.
Now, simplifying the right side:
StartFraction 5 Over y + 4 EndFraction - StartFraction 7y + 8 Over y^2 -16 EndFraction = StartFraction 5(y - 4) Over (y^2 - 16) + StartFraction 7y + 8 Over y^2 - 16 EndFraction.
Combining both sides of the equation:
StartFraction -8 + 8y Over y EndFraction = StartFraction 5(y - 4) Over (y^2 - 16) + StartFraction 7y + 8 Over y^2 - 16 EndFraction.
Multiplying through by y(y^2 - 16) to eliminate the denominators:
-8(y^2 - 16) + 8y(y^2 - 16) = 5y(y - 4) + (7y + 8)y.
Expanding and simplifying:
-8y^2 + 128 + 8y^3 - 128y = 5y^2 - 20y + 7y^2 - 28y + 8y^2.
Combining like terms:
8y^3 - 8y^2 - 34y + 128 = 8y^3 + 22y^2 - 48y.
Subtracting (8y^3 + 22y^2 - 48y) from both sides:
0 = 30y^2 - 14y - 128.
Now, we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
y = (-(-14) ± √((-14)^2 - 4(30)(-128))) / (2(30)).
Simplifying:
y = (14 ± √(196 + 15360)) / 60.
y = (14 ± √15556) / 60.
y ≈ (14 ± 124.64) / 60.
This gives us two possible solutions:
y ≈ (14 + 124.64) / 60 ≈ 2.399.
y ≈ (14 - 124.64) / 60 ≈ -1.177.
Thus, the solutions to the equation are y ≈ 2.399 and y ≈ -1.177.