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

Options

A. `(x-c)/((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 \(\frac{a-c}{(a-b)(x-a)} + \frac{b-c}{(b-a)(x-b)}\), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{a-c}{(a-b)(x-a)} + \frac{b-c}{(b-a)(x-b)} \]

### Step 2: Find a common denominator
The common denominator for the two fractions is \((a-b)(b-a)(x-a)(x-b)\). Notice that \((b-a) = -(a-b)\), so we can rewrite the second term:
\[ \frac{b-c}{(b-a)(x-b)} = \frac{-(b-c)}{(a-b)(x-b)} \]

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

### Step 4: Expand the numerator
Now we expand the numerator:
\[ (a-c)(x-b) - (b-c)(x-a) = ax - ab - cx + bc - bx + ac + cx - ac \]
This simplifies to:
\[ ax - bx - ab + bc \]

### Step 5: Factor the numerator
We can factor the numerator:
\[ (a-b)x + (bc - ab) \]

### Step 6: Substitute back into the expression
Now, we substitute back into the expression:
\[ \frac{(a-b)x + (bc - ab)}{(a-b)(b-a)(x-a)(x-b)} \]

### Step 7: Simplify the expression
Since \((a-b)\) is common in the numerator and denominator, we can cancel it out:
\[ \frac{x + \frac{bc - ab}{a-b}}{(b-a)(x-a)(x-b)} \]

### Step 8: Final expression
The final expression simplifies to:
\[ \frac{x - c}{(x-a)(x-b)} \]

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

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