Question

solve for t. if there are multiple solutions, enter them as a list of values separated by commas. if there are no solutions, type none.\\(\frac{1}{t - 5} + \frac{2}{t + 5} = \frac{5}{t^2 - 25}\\)\\(t = \\)\
enter any extraneous solutions. if there are multiple extraneous solutions, enter them as a list separated by a comma. if there are no extraneous solutions, enter none.\\(t = \\)

Explanation:

Step1: Factor the denominator

Notice that \( t^2 - 25=(t - 5)(t + 5) \), so the equation is \(\frac{1}{t - 5}+\frac{2}{t + 5}=\frac{5}{(t - 5)(t + 5)}\).

Step2: Multiply through by \((t - 5)(t + 5)\)

To eliminate the denominators, multiply each term by \((t - 5)(t + 5)\):
\((t + 5)+2(t - 5)=5\)

Step3: Expand and simplify

Expand the left - hand side: \(t + 5+2t-10 = 5\).
Combine like terms: \(3t-5 = 5\).

Step4: Solve for \(t\)

Add 5 to both sides: \(3t=5 + 5=10\).
Divide both sides by 3: \(t=\frac{10}{3}\).

Step5: Check for extraneous solutions

The original equation has restrictions \(t
eq5\) and \(t
eq - 5\) (because these values make the denominators zero).
Substitute \(t = \frac{10}{3}\) into the original equation:
Left - hand side: \(\frac{1}{ \frac{10}{3}-5}+\frac{2}{\frac{10}{3}+5}=\frac{1}{\frac{10 - 15}{3}}+\frac{2}{\frac{10 + 15}{3}}=\frac{1}{-\frac{5}{3}}+\frac{2}{\frac{25}{3}}=-\frac{3}{5}+\frac{6}{25}=\frac{-15 + 6}{25}=-\frac{9}{25}\)
Right - hand side: \(\frac{5}{(\frac{10}{3})^2-25}=\frac{5}{\frac{100}{9}-25}=\frac{5}{\frac{100 - 225}{9}}=\frac{5}{-\frac{125}{9}}=-\frac{9}{25}\)
Since \(t=\frac{10}{3}\) does not make the denominator zero and satisfies the equation, there are no extraneous solutions.

Answer:

(for the first part, solving for \(t\)):
\(\frac{10}{3}\)