Answer:
To solve the equation, let's simplify each side separately and then combine like terms:
Left side:
\[3x + 14 - 5x + 12 - x + 38 = (3x - 5x - x) + (14 + 12 + 38) = -3x + 64\]
Right side:
\[3 - 2x + 13 - 11 = (3 - 11) + (-2x + 13) = -8 - 2x + 13 = -2x + 5\]
Now, we rewrite the equation:
\[-3x + 64 = -2x + 5\]
To isolate \(x\), let's move all terms involving \(x\) to one side:
\[-3x + 2x = 5 - 64\]
\[(-3 + 2)x = -59\]
\[x = \frac{-59}{-1}\]
\[x = 59\]
Now, we check the solution by substituting \(x = 59\) into the original equation:
\[ \frac{3(59) + 14 - 5(59) + 12 - 59 + 38}{4} = \frac{3 - 2(59) + 13 - 11}{5} \]
\[ \frac{177 + 14 - 295 + 12 - 59 + 38}{4} = \frac{3 - 118 + 13 - 11}{5} \]
\[ \frac{-113}{4} = \frac{-113}{5} \]
Since both sides of the equation have the same value, \(x = 59\) is the correct solution.
