If `(a)/(x-1) + (4b)/(y-2b) + (9c)/(z-3c)= 6-a` find `(ax)/(x-1) + (2y)/(y-2b) + (3z)/(z-3c)`= ?

Answer

To solve the problem, we start with the given equation:

\[ \frac{a}{x-1} + \frac{4b}{y-2b} + \frac{9c}{z-3c} = 6 - a \]

We need to find the value of:

\[ \frac{ax}{x-1} + \frac{2y}{y-2b} + \frac{3z}{z-3c} \]

Let's denote the expression we want to find as \( T \):

\[ T = \frac{ax}{x-1} + \frac{2y}{y-2b} + \frac{3z}{z-3c} \]

### Step 1: Rewrite \( T \)

We can rewrite \( T \) in terms of the original equation. Notice that:

\[ \frac{ax}{x-1} = a + \frac{a}{x-1} \]

\[ \frac{2y}{y-2b} = 2 + \frac{4b}{y-2b} \]

\[ \frac{3z}{z-3c} = 3 + \frac{9c}{z-3c} \]

Thus, we can express \( T \) as:

\[ T = \left( a + \frac{a}{x-1} \right) + \left( 2 + \frac{4b}{y-2b} \right) + \left( 3 + \frac{9c}{z-3c} \right) \]

### Step 2: Combine the terms

Combining the constant terms gives:

\[ T = a + 2 + 3 + \frac{a}{x-1} + \frac{4b}{y-2b} + \frac{9c}{z-3c} \]

This simplifies to:

\[ T = a + 5 + \left( \frac{a}{x-1} + \frac{4b}{y-2b} + \frac{9c}{z-3c} \right) \]

### Step 3: Substitute the original equation

From the original equation, we know:

\[ \frac{a}{x-1} + \frac{4b}{y-2b} + \frac{9c}{z-3c} = 6 - a \]

Substituting this into our expression for \( T \):

\[ T = a + 5 + (6 - a) \]

### Step 4: Simplify

Now, simplifying gives:

\[ T = a + 5 + 6 - a = 11 \]

### Final Result

Thus, the value of \( T \) is:

\[ \boxed{11} \]