Solving The Equation (y/(y-4))-(4/(y+4))=(32/(y^2-16)) A Step-by-Step Guide

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1. Determining the Domain

Before diving into the algebraic manipulations, it's crucial to determine the domain of the equation. The domain represents all possible values of y for which the equation is defined. In rational equations, we need to exclude any values that would make the denominator equal to zero, as division by zero is undefined. Our equation is:

yy−4−4y+4=32y2−16\frac{y}{y-4}-\frac{4}{y+4}=\frac{32}{y^2-16}

We have three denominators: y - 4, y + 4, and y² - 16. Setting each of these equal to zero will help us find the values to exclude:

  • y - 4 = 0 => y = 4
  • y + 4 = 0 => y = -4
  • y² - 16 = 0 => (y - 4)(y + 4) = 0 => y = 4 or y = -4

Thus, the domain of our equation is all real numbers except 4 and -4. This means that if we find solutions of y = 4 or y = -4, they will be extraneous solutions and must be discarded. Understanding the domain restrictions is a critical first step in solving rational equations, ensuring that the final solutions are valid.

2. Clearing the Fractions

To solve the equation, the next step is to eliminate the fractions. This can be achieved by multiplying both sides of the equation by the least common denominator (LCD) of all the fractions. In this case, the denominators are y - 4, y + 4, and y² - 16. Notice that y² - 16 can be factored as (y - 4)(y + 4). Therefore, the LCD is (y - 4)(y + 4).

Multiplying both sides of the equation by the LCD, we get:

(y - 4)(y + 4) * [$\frac{y}{y-4}-\frac{4}{y+4}$] = (y - 4)(y + 4) * [$\frac{32}{y^2-16}$]

Distribute the LCD to each term:

(y - 4)(y + 4) * $\frac{y}{y-4}$ - (y - 4)(y + 4) * $\frac{4}{y+4}$ = (y - 4)(y + 4) * $\frac{32}{(y-4)(y+4)}$

Now, we can cancel out common factors in each term:

( y + 4) * y - (y - 4) * 4 = 32

This simplifies the equation by eliminating the fractions, making it easier to solve. The next step involves expanding and simplifying the resulting equation to solve for y. Clearing fractions is a common and effective technique in solving rational equations, transforming the problem into a more manageable form.

3. Solving the Simplified Equation

After clearing the fractions, we now have a simplified equation:

( y + 4) * y - (y - 4) * 4 = 32

First, expand the terms:

y² + 4y - 4y + 16 = 32

Next, combine like terms:

y² + 16 = 32

Now, subtract 32 from both sides to set the equation to zero:

y² - 16 = 0

This is a quadratic equation, which can be solved by factoring, completing the square, or using the quadratic formula. In this case, factoring is the most straightforward approach. We recognize that y² - 16 is a difference of squares, which factors as:

(y - 4)(y + 4) = 0

Setting each factor equal to zero gives us the potential solutions:

  • y - 4 = 0 => y = 4
  • y + 4 = 0 => y = -4

However, it's crucial to remember the domain restrictions we identified earlier. We found that y cannot be 4 or -4 because these values make the original denominators zero. Therefore, both y = 4 and y = -4 are extraneous solutions. The process of solving the simplified equation is a critical step, but it's equally important to consider domain restrictions to ensure the validity of the solutions.

4. Checking for Extraneous Solutions

As discussed earlier, it is essential to check for extraneous solutions in rational equations. Extraneous solutions are values that satisfy the simplified equation but not the original equation due to domain restrictions. In our case, we found potential solutions y = 4 and y = -4. However, we determined earlier that the domain of the original equation excludes these values.

Let's verify this by substituting these values back into the original equation:

yy−4−4y+4=32y2−16\frac{y}{y-4}-\frac{4}{y+4}=\frac{32}{y^2-16}

For y = 4:

44−4−44+4=3242−16\frac{4}{4-4}-\frac{4}{4+4}=\frac{32}{4^2-16}

This simplifies to:

40−48=320\frac{4}{0}-\frac{4}{8}=\frac{32}{0}

Since we have division by zero, y = 4 is indeed an extraneous solution.

For y = -4:

−4−4−4−4−4+4=32(−4)2−16\frac{-4}{-4-4}-\frac{4}{-4+4}=\frac{32}{(-4)^2-16}

This simplifies to:

−4−8−40=320\frac{-4}{-8}-\frac{4}{0}=\frac{32}{0}

Again, we encounter division by zero, confirming that y = -4 is also an extraneous solution. Therefore, after checking for extraneous solutions, we conclude that the original equation has no valid solutions. This step is crucial in solving rational equations, as it ensures that the obtained solutions are mathematically sound and consistent with the original problem.

5. Conclusion

In summary, to solve the equation $\frac{y}{y-4}-\frac{4}{y+4}=\frac{32}{y^2-16}$, we followed a systematic approach:

  1. Determined the domain: Identified that y cannot be 4 or -4.
  2. Cleared the fractions: Multiplied both sides by the LCD (y - 4)(y + 4).
  3. Solved the simplified equation: Factored the quadratic equation and found potential solutions y = 4 and y = -4.
  4. Checked for extraneous solutions: Verified that both potential solutions were extraneous because they resulted in division by zero in the original equation.

Therefore, the final answer is that the equation has no solution. This comprehensive step-by-step guide illustrates the importance of each stage in solving rational equations, emphasizing the necessity of identifying domain restrictions and checking for extraneous solutions. By following these steps, you can confidently tackle similar problems and ensure accurate results. Understanding the nuances of rational equations, including domain restrictions and extraneous solutions, is crucial for mastering algebraic problem-solving.