Question

equations
what is the solution to the equation \\(\frac{1}{h - 5} + \frac{2}{h + 5} = \frac{16}{h^2 - 25}\\)?
\\(h = 5\\) \\(h = 7\\) \\(h = \frac{21}{2}\\)
\\(h = \frac{11}{3}\\)

Explanation:

Step1: Factor the denominator

Notice that \(h^2 - 25\) is a difference of squares, so \(h^2 - 25=(h - 5)(h + 5)\). The equation becomes \(\frac{1}{h - 5}+\frac{2}{h + 5}=\frac{16}{(h - 5)(h + 5)}\).

Step2: Multiply through by the LCD

The least common denominator (LCD) of the fractions is \((h - 5)(h + 5)\). Multiply each term by \((h - 5)(h + 5)\) to eliminate the denominators:

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Step3: Simplify and solve for \(h\)

Expand the left - hand side: \(h + 5+2h-10 = 16\).
Combine like terms: \(3h-5 = 16\).
Add 5 to both sides: \(3h=16 + 5=21\).
Divide both sides by 3: \(h = 7\).

We need to check for extraneous solutions. If \(h = 5\), the denominators \(h - 5\) and \(h^2-25\) are zero, so \(h = 5\) is not a valid solution. For \(h = 7\), the denominators \(h-5=2\), \(h + 5 = 12\) and \(h^2-25=49 - 25 = 24\) are all non - zero.

Answer:

\(h = 7\)