If `( 11 - 13 x)/( 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 will simplify each term and find the value of \(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\).

### Step 1: Rewrite the equation
We can rewrite each term in the equation:

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

Substituting these back into the equation gives us:

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

### Step 2: Combine like terms
Now we can combine the constant terms and the fractions:

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

### Step 3: Isolate the fraction terms
Next, we add 39 to both sides of the equation:

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

### Step 4: 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 5: Solve for \(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\)
Now, we divide both sides by 11:

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

Thus, the value of \(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\) is **4**.