If `(a - b)(a + b) = 7 xx 13`, then which of the following pairs could be the value of a and b. respectively?

Options

A. `-10,3`
B. 10,3
C. `-3, -10`
D. `-10,-3`

Answer

To solve the equation \((a - b)(a + b) = 7 \times 13\), we can follow these steps:

### Step 1: Calculate the right-hand side
First, we calculate the right-hand side of the equation:
\[ 7 \times 13 = 91 \]
So, we rewrite the equation:
\[ (a - b)(a + b) = 91 \]

### Step 2: Identify possible pairs
Since \(91\) can be expressed as a product of pairs of factors, we can consider the following pairs:
1. \( (a - b, a + b) = (7, 13) \)
2. \( (a - b, a + b) = (-7, -13) \)
3. \( (a - b, a + b) = (13, 7) \)
4. \( (a - b, a + b) = (-13, -7) \)

### Step 3: Solve each case

#### Case 1: \(a - b = 7\) and \(a + b = 13\)
Add the two equations:
\[ (a - b) + (a + b) = 7 + 13 \implies 2a = 20 \implies a = 10 \]
Substituting \(a = 10\) into \(a + b = 13\):
\[ 10 + b = 13 \implies b = 3 \]
So, one solution is \((a, b) = (10, 3)\).

#### Case 2: \(a - b = -7\) and \(a + b = -13\)
Add the two equations:
\[ (a - b) + (a + b) = -7 - 13 \implies 2a = -20 \implies a = -10 \]
Substituting \(a = -10\) into \(a + b = -13\):
\[ -10 + b = -13 \implies b = -3 \]
So, another solution is \((a, b) = (-10, -3)\).

#### Case 3: \(a - b = 13\) and \(a + b = 7\)
Add the two equations:
\[ (a - b) + (a + b) = 13 + 7 \implies 2a = 20 \implies a = 10 \]
Substituting \(a = 10\) into \(a + b = 7\):
\[ 10 + b = 7 \implies b = -3 \]
So, another solution is \((a, b) = (10, -3)\).

#### Case 4: \(a - b = -13\) and \(a + b = -7\)
Add the two equations:
\[ (a - b) + (a + b) = -13 - 7 \implies 2a = -20 \implies a = -10 \]
Substituting \(a = -10\) into \(a + b = -7\):
\[ -10 + b = -7 \implies b = 3 \]
So, another solution is \((a, b) = (-10, 3)\).

### Summary of Solutions
The possible pairs of \((a, b)\) are:
1. \((10, 3)\)
2. \((-10, -3)\)
3. \((10, -3)\)
4. \((-10, 3)\)