If ` (( x + y))/(z) = 2 ` then what is the value of ` [(y)/((y - z))] + [(x)/( ( x-z))]` ?

Options

A. 0
B. 1
C. 2
D. `-1`

Answer

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

\[ \frac{x + y}{z} = 2 \]

From this equation, we can express \(x + y\) in terms of \(z\):

\[ x + y = 2z \]

Next, we need to find the value of the expression:

\[ \frac{y}{y - z} + \frac{x}{x - z} \]

We will substitute \(y\) and \(x\) in terms of \(z\) using the equation \(x + y = 2z\).

1. **Step 1: Express \(y\) in terms of \(x\) and \(z\)**

From \(x + y = 2z\), we can express \(y\) as:

\[ y = 2z - x \]

2. **Step 2: Substitute \(y\) into the expression**

Now, we substitute \(y\) into the expression:

\[ \frac{y}{y - z} + \frac{x}{x - z} = \frac{2z - x}{(2z - x) - z} + \frac{x}{x - z} \]

Simplifying the first term:

\[ = \frac{2z - x}{z - x} + \frac{x}{x - z} \]

3. **Step 3: Simplify the expression**

The first term can be rewritten as:

\[ = \frac{2z - x}{z - x} + \frac{x}{x - z} \]

Notice that \(\frac{x}{x - z} = -\frac{x}{z - x}\). Thus, we can combine the two fractions:

\[ = \frac{2z - x - x}{z - x} = \frac{2z - 2x}{z - x} \]

This simplifies to:

\[ = 2 \cdot \frac{z - x}{z - x} = 2 \]

Thus, the value of the expression \(\frac{y}{y - z} + \frac{x}{x - z}\) is:

\[ \boxed{2} \]