Questions: Solve: 2/(b-4) + 7/(b+4) = 16/(b^2-16) b=

Transcript text: Solve: \[ \begin{array}{l} \frac{2}{b-4}+\frac{7}{b+4}=\frac{16}{b^{2}-16} \\ b= \end{array} \] $\square$ (Enter your answers, separated by commas, or type DNE if no solution exists) Check Answer

Solution

Solution Steps

To solve the given equation, we first recognize that the right-hand side can be rewritten using the difference of squares: $ b^2 - 16 = (b-4)(b+4) $. This allows us to combine the fractions on the left-hand side over a common denominator of $ (b-4)(b+4) $. Once the fractions are combined, we equate the numerators and solve the resulting equation for $ b $.

Step 1: Rewrite the Equation

We start with the equation: \[ \frac{2}{b-4} + \frac{7}{b+4} = \frac{16}{b^2 - 16} \] Recognizing that $ b^2 - 16 = (b-4)(b+4) $, we can rewrite the equation as: \[ \frac{2(b+4) + 7(b-4)}{(b-4)(b+4)} = \frac{16}{(b-4)(b+4)} \]

Step 2: Combine the Left Side

Combining the fractions on the left side gives us: \[ \frac{2(b+4) + 7(b-4)}{(b-4)(b+4)} = \frac{2b + 8 + 7b - 28}{(b-4)(b+4)} = \frac{9b - 20}{(b-4)(b+4)} \] Thus, we have: \[ \frac{9b - 20}{(b-4)(b+4)} = \frac{16}{(b-4)(b+4)} \]

Step 3: Equate the Numerators

Since the denominators are the same, we can equate the numerators: \[ 9b - 20 = 16 \]

Step 4: Solve for $ b $

Solving the equation $ 9b - 20 = 16 $: \[ 9b = 36 \implies b = 4 \]

Step 5: Check for Valid Solutions

However, we must check if $ b = 4 $ is valid. Substituting $ b = 4 $ into the original denominators $ b-4 $ and $ b+4 $ shows that $ b-4 = 0 $, which makes the original equation undefined. Therefore, $ b = 4 $ is not a valid solution.