If ` ( 4 x - 3)/( x) + ( 4 y - 3)/( y) + ( 4 z - 3)/( z) = 0` then the value of `(1)/( x) + (1)/( y) + (1)/( z)` is

Options

A. 9
B. 3
C. 4
D. 6

Answer

To solve the equation

\[ \frac{4x - 3}{x} + \frac{4y - 3}{y} + \frac{4z - 3}{z} = 0, \]

we will follow these steps:

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

\[ \frac{4x - 3}{x} = 4 - \frac{3}{x}, \quad \frac{4y - 3}{y} = 4 - \frac{3}{y}, \quad \frac{4z - 3}{z} = 4 - \frac{3}{z}. \]

Substituting these into the equation gives:

\[ \left(4 - \frac{3}{x}\right) + \left(4 - \frac{3}{y}\right) + \left(4 - \frac{3}{z}\right) = 0. \]

### Step 2: Combine the terms
Combining the constant terms and the fractions, we have:

\[ 12 - \left(\frac{3}{x} + \frac{3}{y} + \frac{3}{z}\right) = 0. \]

### Step 3: Isolate the fraction sum
Now, isolate the sum of the fractions:

\[ 12 = \frac{3}{x} + \frac{3}{y} + \frac{3}{z}. \]

### Step 4: Factor out the common term
We can factor out 3 from the right side:

\[ 12 = 3\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right). \]

### Step 5: Solve for the desired expression
Now, divide both sides by 3:

\[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{12}{3} = 4. \]

### Final Answer
Thus, the value of

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

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