If `(11- 13x)/(x) + (11- 13 y)/(y) + (11- 13 z)/(z) =5,` then what is the value of `1/x + 1/y + 1/z` ?

Options

A. 1
B. `13/11`
C. `13/5`
D. `4`

Answer

To solve the equation

\[ \frac{11 - 13x}{x} + \frac{11 - 13y}{y} + \frac{11 - 13z}{z} = 5, \]

we can start by simplifying each term in the equation.

### Step 1: Simplify each term

We can rewrite each term in the equation:

\[ \frac{11 - 13x}{x} = \frac{11}{x} - 13, \]
\[ \frac{11 - 13y}{y} = \frac{11}{y} - 13, \]
\[ \frac{11 - 13z}{z} = \frac{11}{z} - 13. \]

### Step 2: Substitute back into the equation

Now, substituting these simplified terms back into the original equation gives us:

\[ \left(\frac{11}{x} - 13\right) + \left(\frac{11}{y} - 13\right) + \left(\frac{11}{z} - 13\right) = 5. \]

### Step 3: Combine like terms

Combining the constant terms and the fractions, we have:

\[ \frac{11}{x} + \frac{11}{y} + \frac{11}{z} - 39 = 5. \]

### Step 4: Isolate the fraction terms

Next, we can isolate the fraction terms by adding 39 to both sides:

\[ \frac{11}{x} + \frac{11}{y} + \frac{11}{z} = 5 + 39, \]
\[ \frac{11}{x} + \frac{11}{y} + \frac{11}{z} = 44. \]

### Step 5: Factor out the common term

Now, we can factor out 11 from the left side:

\[ 11\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) = 44. \]

### Step 6: Solve for \( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \)

To find \( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \), we divide both sides by 11:

\[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{44}{11} = 4. \]

### Final Answer

Thus, the value of \( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \) is

\[ \boxed{4}. \]