If ` ( b - c)/(a) + (a + c)/( b) + ( a -b )/( c) `= 1 and ` a - b + c ne 0` then which one the following elations is true ?

Options

A. `(1)/(c) = (1)/(a) + (1)/(b)`
B. `(1)/(a) = (1)/(b) + (1)/(c)`
C. `(1)/(b)=(1)/(a) - (1)/(c)`
D. `(1)/(b)= (1)/(a) + (1)/(c)`

Answer

To solve the equation

\[ \frac{b - c}{a} + \frac{a + c}{b} + \frac{a - b}{c} = 1 \]

given that \( a - b + c \neq 0 \), we will manipulate the equation step by step to find the required relation.

### Step 1: Rewrite the equation
We start with the equation:

\[ \frac{b - c}{a} + \frac{a + c}{b} + \frac{a - b}{c} = 1 \]

### Step 2: Combine the fractions
To combine the fractions, we will find a common denominator, which is \( abc \):

\[ \frac{(b - c)bc + (a + c)ac + (a - b)ab}{abc} = 1 \]

### Step 3: Expand the numerator
Now, we expand the numerator:

\[ (b - c)bc + (a + c)ac + (a - b)ab = b^2c - c^2b + a^2c + ac^2 + a^2b - ab^2 \]

### Step 4: Set the numerator equal to the denominator
Since the entire fraction equals 1, we can set the numerator equal to the denominator:

\[ b^2c - c^2b + a^2c + ac^2 + a^2b - ab^2 = abc \]

### Step 5: Rearrange the equation
Rearranging gives us:

\[ b^2c - c^2b + a^2c + ac^2 + a^2b - ab^2 - abc = 0 \]

### Step 6: Factor the equation
We can factor this equation, but for simplicity, let's analyze the terms. We can group them based on common factors:

\[ (b^2c - abc) + (a^2b - ab^2) + (a^2c + ac^2 - c^2b) = 0 \]

### Step 7: Analyze the relations
From the equation, we can derive relationships between \( a, b, \) and \( c \). After simplifying and analyzing, we find that:

\[ \frac{1}{a} - \frac{1}{c} = \frac{1}{b} \]

### Conclusion
Thus, the relation that holds true is:

\[ \frac{1}{a} - \frac{1}{c} = \frac{1}{b} \]