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.)
\[
\frac{3}{x-4}=\frac{1}{x-1}+\frac{7}{(x-1)(x-4)}
\]
Final Answer: The solution to the equation is \(\boxed{3}\).
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 \(\boxed{3}\).