If `(x+1)/((x-a)(x-3))=2/(x-a)+b/(x-3), " then "(a, b)=`

Options

A. (7, -1)
B. (-4, 1)
C. (4, 1)
D. (-4, -1)

Answer

To solve the equation \(\frac{x+1}{(x-a)(x-3)} = \frac{2}{x-a} + \frac{b}{x-3}\), we will follow these steps:

### Step 1: Write the equation
We start with the given equation:
\[ \frac{x+1}{(x-a)(x-3)} = \frac{2}{x-a} + \frac{b}{x-3} \]

### Step 2: Find a common denominator
The common denominator for the right-hand side is \((x-a)(x-3)\). We rewrite the right side:
\[ \frac{2(x-3) + b(x-a)}{(x-a)(x-3)} \]
So, we have:
\[ \frac{x+1}{(x-a)(x-3)} = \frac{2(x-3) + b(x-a)}{(x-a)(x-3)} \]

### Step 3: Cross-multiply
Since the denominators are the same, we can equate the numerators:
\[ x + 1 = 2(x - 3) + b(x - a) \]

### Step 4: Expand the right-hand side
Expanding the right side gives:
\[ x + 1 = 2x - 6 + bx - ab \]

### Step 5: Rearrange the equation
Rearranging the equation, we get:
\[ x + 1 = (2 + b)x - (6 + ab) \]

### Step 6: Compare coefficients
Now, we compare the coefficients of \(x\) and the constant terms on both sides:
1. Coefficient of \(x\):
\[ 1 = 2 + b \implies b = 1 - 2 = -1 \]
2. Constant terms:
\[ 1 = -6 - ab \]
Substituting \(b = -1\):
\[ 1 = -6 - a(-1) \implies 1 = -6 + a \implies a = 1 + 6 = 7 \]

### Step 7: Final result
Thus, we find:
\[ (a, b) = (7, -1) \]