Solve $ \frac{5}{a - 4} + \frac{2}{a + 1} = \frac{-31}{a^2 - 3a - 4} $ and Check for Extraneous Solution

Question

Solve $ \frac{5}{a - 4} + \frac{2}{a + 1} = \frac{-31}{a^2 - 3a - 4} $ and check for extraneous solution.

Accepted Answer

C

Answer

Factor right denominator: $ a^2 - 3a - 4 = (a - 4)(a + 1) $ Multiply by LCD $ (a - 4)(a + 1) $: $ 5(a + 1) + 2(a - 4) = -31 $ Expand and simplify: $ 5a + 5 + 2a - 8 = -31 $ → $ 7a - 3 = -31 $ → $ 7a = -28 $ → $ a = -4 $ Check $ a = -4 $: Substitute into original equation; both sides equal (no extraneous solution)