What is the solution of $\frac{-8}{2 y-8}=\frac{5}{y+4}-\frac{7 y+8}{y^2-16}$ ?
To solve the equation \(\frac{-8}{2y-8} = \frac{5}{y+4} - \frac{7y+8}{y^2-16}\), we first analyze the denominators to determine the domain restrictions. The denominators are \(2y-8 = 2(y-4)\), \(y+4\), and \(y^2-16 = (y-4)(y+4)\). For the equation to be defined, these denominators cannot be zero. Therefore, \(y eq 4\) and \(y eq -4\).
Next, we identify the least common denominator (LCD) for all fractions, which is \(2(y-4)(y+4)\). To clear the denominators, we multiply both sides of the equation by this LCD:
\[2(y-4)(y+4) \cdot \left(\frac{-8}{2(y-4)}\right) = 2(y-4)(y+4) \cdot \left(\frac{5}{y+4} - \frac{7y+8}{(y-4)(y+4)}\right)\]
This simplifies to:
\[-8(y+4) = 5 \cdot 2(y-4) - (7y+8) \cdot 2\]
Now, we expand both sides of the equation:
\[-8y - 32 = 10(y-4) - 2(7y+8)\]
\[-8y - 32 = 10y - 40 - 14y - 16\]
Combining like terms on the right side:
\[-8y - 32 = -4y - 56\]
To isolate \(y\), add \(8y\) to both sides and add \(56\) to both sides:
\[-32 + 56 = -4y + 8y\]
\[24 = 4y\]
Dividing by \(4\), we get \(y = 6\). Since \(6\) is not one of the restricted values (\(4\) or \(-4\)), this is a valid solution. Looking at the options provided, \(y = 6\) is the correct choice.
Solving rational equations is essential in fields like engineering and physics when modeling systems where rates are involved. For example, if two workers or machines have different processing speeds and you need to find the total time required to complete a task working together, you set up a rational equation of the form \(\frac{1}{t_1} + \frac{1}{t_2} = \frac{1}{T}\). Similar techniques ensure that constraints are respected and tasks are completed efficiently.