What is the solution of $\frac{-8}{2 y-8}=\frac{5}{y+4}-\frac{7 y+8}{y^2-16}$?
A. $y=-4$
B. $y=-2$
C. $y=4$
D. $y=6
Answer (2)
Factor the denominators of the rational expressions.
Multiply both sides of the equation by the least common denominator to eliminate fractions.
Simplify and solve the resulting equation for y .
Verify the solution by substituting it back into the original equation to ensure it is valid. The solution is y = 6 .
Explanation
Understanding the Problem We are given the equation 2 y − 8 − 8 = y + 4 5 − y 2 − 16 7 y + 8 and we need to find the value of y that satisfies this equation. The possible values of y are y = − 4 , y = − 2 , y = 4 , y = 6 . Note that y 2 − 16 = ( y − 4 ) ( y + 4 ) and 2 y − 8 = 2 ( y − 4 ) . We must exclude y = 4 and y = − 4 from the possible solutions, since they make the denominators zero.
Factoring the Denominators First, let's factor the denominators to get a better understanding of the equation: 2 y − 8 = 2 ( y − 4 ) and y 2 − 16 = ( y − 4 ) ( y + 4 ) . So, we can rewrite the equation as 2 ( y − 4 ) − 8 = y + 4 5 − ( y − 4 ) ( y + 4 ) 7 y + 8 .
Eliminating the Denominators Now, let's multiply both sides of the equation by 2 ( y − 4 ) ( y + 4 ) to eliminate the denominators. This gives us − 8 ( y + 4 ) = 5 ( 2 ( y − 4 )) − 2 ( 7 y + 8 ) .
Simplifying the Equation Next, we simplify the equation: − 8 y − 32 = 10 y − 40 − 14 y − 16 .
Combining Like Terms Now, let's combine like terms: − 8 y − 32 = − 4 y − 56 .
Solving for y Now, we solve for y : 4 y = 24 , so y = 6 .
Checking the Solution Finally, we check if y = 6 is a valid solution by substituting it into the original equation. Since y = 6 is not 4 or -4, it is a valid solution. Therefore, the solution is y = 6 .
Examples
When solving problems involving rates or ratios, such as comparing the speeds of two objects or determining the concentration of a mixture, you might encounter rational equations similar to the one we just solved. For instance, if you are analyzing the flow rates of two pipes filling a tank, you could set up an equation where the rates are expressed as fractions, and the denominators involve variables representing time or volume. Solving such equations helps you find the time it takes to fill the tank or the volume filled by each pipe in a given time. Understanding how to manipulate and solve these equations is crucial for making accurate predictions and optimizing processes in various real-world scenarios.
The solution to the equation 2 y − 8 − 8 = y + 4 5 − y 2 − 16 7 y + 8 is y = 6 . This was found by simplifying the equation and solving for y , ensuring the solution does not make any denominators zero. Therefore, the correct answer is y = 6 .
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