The value of `a/((a-b)(x-a))+b/((b-a)(x-b))` is

Options

A. `x/((x-a)(x-b))`
B. `(x-a)/((x-b)(x-c))`
C. `(x - c)/((x - b)(a-x))`
D. None of these

Answer

To solve the expression \( \frac{a}{(a-b)(x-a)} + \frac{b}{(b-a)(x-b)} \), we can follow these steps:

### Step 1: Identify the common denominator
The denominators of the two fractions are \( (a-b)(x-a) \) and \( (b-a)(x-b) \). Notice that \( (b-a) = -(a-b) \). Therefore, we can rewrite the second denominator as:
\[ (b-a)(x-b) = -(a-b)(x-b) \]
Thus, the common denominator for both fractions is:
\[ (a-b)(x-a)(x-b) \]

### Step 2: Rewrite each fraction with the common denominator
Now we can rewrite each fraction with the common denominator:
\[ \frac{a}{(a-b)(x-a)} = \frac{a(x-b)}{(a-b)(x-a)(x-b)} \]
\[ \frac{b}{(b-a)(x-b)} = \frac{-b(x-a)}{(a-b)(x-a)(x-b)} \]

### Step 3: Combine the fractions
Now we can combine the two fractions:
\[ \frac{a(x-b) - b(x-a)}{(a-b)(x-a)(x-b)} \]

### Step 4: Simplify the numerator
Now let's simplify the numerator:
\[ a(x-b) - b(x-a) = ax - ab - bx + ba = ax - bx = (a-b)x \]

### Step 5: Write the final expression
Now we can substitute the simplified numerator back into the fraction:
\[ \frac{(a-b)x}{(a-b)(x-a)(x-b)} \]

### Step 6: Cancel common factors
Since \( (a-b) \) is common in both the numerator and denominator, we can cancel it (assuming \( a \neq b \)):
\[ \frac{x}{(x-a)(x-b)} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{x}{(x-a)(x-b)} \]