If ` ( 3 x - 1)/( x) + ( 5 y - 1)/( y) + ( 7z - 1)/( z)=0` what is the value of `((1)/( x)+(1)/( y) + (1)/(z))` ?

Options

A. `3`
B. 0
C. 15
D. 21

Answer

To solve the equation

\[ \frac{3x - 1}{x} + \frac{5y - 1}{y} + \frac{7z - 1}{z} = 0, \]

we will simplify each term and find the value of

\[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z}. \]

### Step 1: Rewrite each term
We can rewrite each fraction as follows:

\[ \frac{3x - 1}{x} = 3 - \frac{1}{x}, \quad \frac{5y - 1}{y} = 5 - \frac{1}{y}, \quad \frac{7z - 1}{z} = 7 - \frac{1}{z}. \]

### Step 2: Substitute back into the equation
Substituting these into the original equation gives:

\[ \left(3 - \frac{1}{x}\right) + \left(5 - \frac{1}{y}\right) + \left(7 - \frac{1}{z}\right) = 0. \]

### Step 3: Combine like terms
Combining the constant terms and the fractions results in:

\[ 3 + 5 + 7 - \left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) = 0. \]

### Step 4: Simplify the equation
Adding the constants gives:

\[ 15 - \left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) = 0. \]

### Step 5: Solve for \(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\)
Rearranging the equation, we find:

\[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 15. \]

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

\[ \boxed{15}. \]