Given `x+ (1)/(y)=1 and y + (1)/(z)=1` find
`(x+y+z) + ((1)/(x) + (1)/(y) + (1)/(z))=`?

Answer

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

1. \( x + \frac{1}{y} = 1 \)
2. \( y + \frac{1}{z} = 1 \)

We need to find the value of:

\[ (x + y + z) + \left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) \]

### Step 1: Express \( x \) in terms of \( y \)

From the first equation, we can isolate \( x \):

\[ x = 1 - \frac{1}{y} \]

### Step 2: Express \( z \) in terms of \( y \)

From the second equation, we can isolate \( z \):

\[ \frac{1}{z} = 1 - y \implies z = \frac{1}{1 - y} \]

### Step 3: Substitute values of \( x \) and \( z \) into the expression

Now we substitute \( x \) and \( z \) into the expression we need to evaluate:

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

### Step 4: Simplify \( x + y + z \)

Combining the terms:

\[ x + y + z = 1 - \frac{1}{y} + y + \frac{1}{1 - y} \]

### Step 5: Find \( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \)

Next, we calculate \( \frac{1}{x} \):

\[ \frac{1}{x} = \frac{1}{1 - \frac{1}{y}} = \frac{y}{y - 1} \]

Now we can find \( \frac{1}{y} \) and \( \frac{1}{z} \):

\[ \frac{1}{y} = \frac{1}{y} \]
\[ \frac{1}{z} = 1 - y \]

So, we have:

\[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{y}{y - 1} + \frac{1}{y} + (1 - y) \]

### Step 6: Combine the two parts

Now we need to combine \( (x + y + z) \) and \( \left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) \):

\[ (x + y + z) + \left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) \]

### Step 7: Substitute \( y = 2 \) for simplification

To simplify our calculations, we can assume \( y = 2 \):

1. From \( x + \frac{1}{2} = 1 \), we find \( x = \frac{1}{2} \).
2. From \( 2 + \frac{1}{z} = 1 \), we find \( z = -1 \).

### Step 8: Calculate the final expression

Now substituting \( x = \frac{1}{2} \), \( y = 2 \), and \( z = -1 \):

\[ x + y + z = \frac{1}{2} + 2 - 1 = \frac{1}{2} + 1 = \frac{3}{2} \]

Now calculate \( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \):

\[ \frac{1}{x} = 2, \quad \frac{1}{y} = \frac{1}{2}, \quad \frac{1}{z} = -1 \]

Thus:

\[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 2 + \frac{1}{2} - 1 = 1 + \frac{1}{2} = \frac{3}{2} \]

### Final Calculation:

Now we combine both parts:

\[ \frac{3}{2} + \frac{3}{2} = 3 \]

Thus, the final answer is:

\[ \boxed{3} \]