Answer:
D) y = 6
Step-by-step explanation:
Add the opposite of the left side:
... 0 = 8/(2y -8) +5/(y +4) -(7y +8)/(y^2 -16)
... 0 = (4(y +4) +5(y -4) -(7y +8)) / (y^2 -16) . . . . use a common denominator
... 0 = 4y +16 +5y -20 -7y -8 . . . . . multiply by (y^2 -16), eliminate parentheses
... 0 = 2y -12 . . . . . collect terms
... 0 = y -6 . . . . . . . divide by 2
... 6 = y . . . . . . . . . add 6
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Comment on the solution
Keeping the expression rational and using a common denominator can avoid introducing the extraneous roots that you get when you multiply by y^2-16 at the beginning of the process. That multiplication creates a polynomial that has the roots of y^2-16 along with the solutions to the original equation.
In the above, we have a step "multiply by (y^2 -16)", but that multiplication does not have the effect of adding extraneous roots. We recognize that that step can only be valid for values of y that are not ±4. The equation itself is not defined for y = ±4. Effectively, we're ignoring the denominator and looking only for values of y that make the numerator zero.
