Solve the rational equation for $x$. Remember to check for values that should be excluded from the solution set. (If there is no solution, enter NO soluTION.) \[ \frac{3}{x-4}=\frac{1}{x-1}+\frac{7}{(x-1)(x-4)} \]

Step 1 ：Find a common denominator for all the fractions in the equation. In this case, the common denominator is \((x-1)(x-4)\).

Step 2 ：Multiply each term by the common denominator to eliminate the fractions.

Step 3 ：Simplify the equation and solve for \(x\).

Step 4 ：Check for values that should be excluded from the solution set. In this case, \(x\) cannot be 1 or 4 because these values would make the denominator of the original equation zero.

Step 5 ：The solution to the equation is \(x = 3\). This value does not make the denominator of the original equation zero, so it is a valid solution.

Step 6 ：Final Answer: The solution to the equation is \(\boxed{3}\).