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{8}{t - 7} + \frac{3}{t + 7} = \frac{9}{t^2 - 49}\\)

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 on the right

Notice that \( t^2 - 49=(t - 7)(t + 7) \), so the equation is \(\frac{8}{t - 7}+\frac{3}{t + 7}=\frac{9}{(t - 7)(t + 7)}\).

Step2: Multiply through by \((t - 7)(t + 7)\) to eliminate denominators

\(8(t + 7)+3(t - 7)=9\).

Step3: Expand the left side

\(8t+56 + 3t-21 = 9\).

Step4: Combine like terms

\(11t+35 = 9\).

Step5: Subtract 35 from both sides

\(11t=9 - 35=-26\).

Step6: Solve for \(t\)

\(t=\frac{-26}{11}\).

Step7: Check for extraneous solutions

The original equation has denominators \(t - 7\), \(t + 7\), and \(t^2 - 49=(t - 7)(t + 7)\). The values \(t = 7\) and \(t=-7\) make the denominators zero, so they are excluded. The solution \(t=\frac{-26}{11}\) does not make any denominator zero, so there are no extraneous solutions.

Answer:

(for \(t\)):
\(-\frac{26}{11}\)