If `(a)/(x-a) + (b)/(y-b) + (c )/(z-c)=2` find `(x)/(x-a) + (y)/(y-b) + (z)/(z-c)`= ?

Answer

To solve the problem, we will follow a systematic approach. We are given the equation:

\[ \frac{a}{x-a} + \frac{b}{y-b} + \frac{c}{z-c} = 2 \]

and we need to find the value of:

\[ \frac{x}{x-a} + \frac{y}{y-b} + \frac{z}{z-c} \]

Let's denote:

\[ T = \frac{x}{x-a} + \frac{y}{y-b} + \frac{z}{z-c} \]

### Step 1: Rewrite the expressions

We can rewrite the expression for \(T\) as follows:

\[ T = \frac{x-a + a}{x-a} + \frac{y-b + b}{y-b} + \frac{z-c + c}{z-c} \]

This simplifies to:

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

### Step 2: Combine the terms

Now, combining the terms gives:

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

### Step 3: Substitute the known value

From the problem statement, we know that:

\[ \frac{a}{x-a} + \frac{b}{y-b} + \frac{c}{z-c} = 2 \]

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

\[ T = 3 + 2 \]

### Step 4: Calculate the final value

Thus, we find:

\[ T = 5 \]

### Conclusion

Therefore, the value of

\[ \frac{x}{x-a} + \frac{y}{y-b} + \frac{z}{z-c} = 5 \]