If ` x + y = 2 z ` then the value jof ` (x)/( x - z) + (z)/( y - z)` iis

Options

A. 1
B. 3
C. `(1)/(2)`
D. 2

Answer

To solve the problem, we need to find the value of the expression \( \frac{x}{x - z} + \frac{z}{y - z} \) given that \( x + y = 2z \).

### Step-by-step Solution:

1. **Start with the given equation**:
\[ x + y = 2z \]

2. **Rearrange the equation to express \( y \) in terms of \( x \) and \( z \)**:
\[ y = 2z - x \]

3. **Substitute \( y \) into the expression**:
We need to evaluate:
\[ \frac{x}{x - z} + \frac{z}{y - z} \]
Substitute \( y = 2z - x \) into the second term:
\[ y - z = (2z - x) - z = z - x \]
Thus, the expression becomes:
\[ \frac{x}{x - z} + \frac{z}{z - x} \]

4. **Find a common denominator**:
The common denominator for the two fractions is \( (x - z)(z - x) \). Note that \( z - x = -(x - z) \), so we can rewrite the expression:
\[ \frac{x(z - x) + z(x - z)}{(x - z)(z - x)} \]

5. **Simplify the numerator**:
Expanding the numerator:
\[ x(z - x) + z(x - z) = xz - x^2 + zx - z^2 = 2xz - x^2 - z^2 \]

6. **Rewrite the expression**:
The expression now looks like:
\[ \frac{2xz - x^2 - z^2}{(x - z)(z - x)} \]
Since \( z - x = -(x - z) \), we can simplify the denominator:
\[ (x - z)(z - x) = -(x - z)^2 \]

7. **Final expression**:
Thus, our expression simplifies to:
\[ \frac{2xz - x^2 - z^2}{-(x - z)^2} \]

8. **Factor the numerator**:
Notice that \( 2xz - x^2 - z^2 = -(x - z)^2 \) (this can be verified by expanding). Therefore, we have:
\[ \frac{-(x - z)^2}{-(x - z)^2} = 1 \]

### Conclusion:
The value of the expression \( \frac{x}{x - z} + \frac{z}{y - z} \) is:
\[ \boxed{1} \]