For which values of a and b will the following pair of linear equations has infinitely many solutions ? `x + 2y = 1` ` (a-b) x + (a + b ) y = a + b -2`

Options

A. `a=3, b=1`
B. `a=1, b=3`
C. `a=1, b=1`
D. `a=3 , b=3`

Explanation

To find the values of \( a \) and \( b \) for which the given pair of linear equations has infinitely many solutions, we will follow these steps:

### Given Equations:
1. \( x + 2y = 1 \)
2. \( (a-b)x + (a+b)y = a + b - 2 \)

### Step 1: Rewrite the equations in standard form
We can rewrite the equations in the form \( Ax + By + C = 0 \).

1. The first equation becomes:
\[ x + 2y - 1 = 0 \]
Here, \( A_1 = 1 \), \( B_1 = 2 \), and \( C_1 = -1 \).

2. The second equation becomes:
\[ (a-b)x + (a+b)y - (a+b-2) = 0 \]
Here, \( A_2 = a-b \), \( B_2 = a+b \), and \( C_2 = -(a+b-2) = -a - b + 2 \).

### Step 2: Condition for infinitely many solutions
For the two equations to have infinitely many solutions, the following condition must hold:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \]

### Step 3: Set up the equations
Substituting the values we have:
\[ \frac{1}{a-b} = \frac{2}{a+b} = \frac{-1}{-a-b+2} \]

### Step 4: Solve the first part of the condition
From \( \frac{1}{a-b} = \frac{2}{a+b} \):
Cross-multiplying gives:
\[ 1 \cdot (a+b) = 2 \cdot (a-b) \]
This simplifies to:
\[ a + b = 2a - 2b \]
Rearranging gives:
\[ a - 3b = 0 \quad \text{(1)} \]
Thus, we have:
\[ a = 3b \]

### Step 5: Solve the second part of the condition
Now, from \( \frac{2}{a+b} = \frac{-1}{-a-b+2} \):
Cross-multiplying gives:
\[ 2 \cdot (-a-b+2) = -1 \cdot (a+b) \]
This simplifies to:
\[ -2a - 2b + 4 = -a - b \]
Rearranging gives:
\[ -a - b + 4 = 0 \quad \text{(2)} \]
Thus, we have:
\[ a + b = 4 \]

### Step 6: Substitute \( a = 3b \) into \( a + b = 4 \)
Substituting \( a = 3b \) into equation (2):
\[ 3b + b = 4 \]
This simplifies to:
\[ 4b = 4 \]
Thus:
\[ b = 1 \]

### Step 7: Find \( a \)
Now substituting \( b = 1 \) back into \( a = 3b \):
\[ a = 3 \cdot 1 = 3 \]

### Conclusion
The values of \( a \) and \( b \) for which the given pair of linear equations has infinitely many solutions are:
\[ \boxed{a = 3, b = 1} \]