What is the solution to the equation y/y-4 -4/y+4=32/y2-16?
The Correct Answer and Explanation is:
To solve the equation
[
\frac{y}{y-4} – \frac{4}{y+4} = \frac{32}{y^2-16},
]
we start by recognizing that (y^2 – 16) can be factored using the difference of squares:
[
y^2 – 16 = (y – 4)(y + 4).
]
This means we can rewrite the equation as:
[
\frac{y}{y-4} – \frac{4}{y+4} = \frac{32}{(y – 4)(y + 4)}.
]
Next, we want a common denominator for the left-hand side of the equation. The common denominator between ((y-4)) and ((y+4)) is ((y-4)(y+4)). We can rewrite each fraction:
[
\frac{y(y + 4)}{(y – 4)(y + 4)} – \frac{4(y – 4)}{(y – 4)(y + 4)} = \frac{32}{(y – 4)(y + 4)}.
]
Combining the fractions on the left gives:
[
\frac{y(y + 4) – 4(y – 4)}{(y – 4)(y + 4)} = \frac{32}{(y – 4)(y + 4)}.
]
Now, simplifying the numerator:
[
y(y + 4) – 4(y – 4) = y^2 + 4y – 4y + 16 = y^2 + 16.
]
So the equation now looks like:
[
\frac{y^2 + 16}{(y – 4)(y + 4)} = \frac{32}{(y – 4)(y + 4)}.
]
Since the denominators are the same and assuming (y \neq 4) and (y \neq -4), we can equate the numerators:
[
y^2 + 16 = 32.
]
Subtracting 32 from both sides gives:
[
y^2 – 16 = 0.
]
Factoring this gives:
[
(y – 4)(y + 4) = 0.
]
Setting each factor to zero yields the solutions (y = 4) and (y = -4). However, since these values cause the original denominators to be zero, they are excluded from the solution set. Thus, the equation has no valid solutions.
In conclusion, upon solving the equation, we found that the potential solutions led to undefined expressions in the original equation, confirming that there are no acceptable values for (y) that satisfy the equation.