For what values of a and b the following pair of linear equations have infinite number of solutions?
`2x+3y=7`
`a(x+y)-b(x-y)=3a+b-2`

Answer

To find the values of \( a \) and \( b \) such that the given pair of linear equations has an infinite number of solutions, we will follow these steps:

### Step 1: Write down the equations
The given equations are:
1. \( 2x + 3y = 7 \) (Equation 1)
2. \( a(x+y) - b(x-y) = 3a + b - 2 \) (Equation 2)

### Step 2: Simplify Equation 2
We will first simplify Equation 2:
\[ a(x+y) - b(x-y) = ax + ay - bx + by = (a-b)x + (a+b)y \]
Thus, Equation 2 can be rewritten as:
\[ (a-b)x + (a+b)y = 3a + b - 2 \]

### Step 3: Identify coefficients
Now we have:
- From Equation 1: \( m_1 = 2, n_1 = 3, p_1 = 7 \)
- From Equation 2: \( m_2 = a-b, n_2 = a+b, p_2 = 3a + b - 2 \)

### Step 4: Set up the condition for infinite solutions
For the two equations to have an infinite number of solutions, the following condition must hold:
\[ \frac{m_1}{m_2} = \frac{n_1}{n_2} = \frac{p_1}{p_2} \]
This gives us the following three equations:
1. \( \frac{2}{a-b} = \frac{3}{a+b} \)
2. \( \frac{2}{a-b} = \frac{7}{3a + b - 2} \)
3. \( \frac{3}{a+b} = \frac{7}{3a + b - 2} \)

### Step 5: Solve the first equation
From the first equation:
\[ 2(a+b) = 3(a-b) \]
Expanding this gives:
\[ 2a + 2b = 3a - 3b \]
Rearranging terms:
\[ 2a - 3a + 2b + 3b = 0 \implies -a + 5b = 0 \implies a = 5b \quad \text{(Equation 4)} \]

### Step 6: Solve the second equation
From the second equation:
\[ 2(3a + b - 2) = 7(a-b) \]
Expanding this gives:
\[ 6a + 2b - 4 = 7a - 7b \]
Rearranging terms:
\[ 6a - 7a + 2b + 7b = 4 \implies -a + 9b = 4 \implies a = 9b - 4 \quad \text{(Equation 5)} \]

### Step 7: Equate Equations 4 and 5
Now we have two expressions for \( a \):
From Equation 4: \( a = 5b \)
From Equation 5: \( a = 9b - 4 \)

Setting them equal:
\[ 5b = 9b - 4 \]
Rearranging gives:
\[ 4 = 9b - 5b \implies 4 = 4b \implies b = 1 \]

### Step 8: Find \( a \)
Substituting \( b = 1 \) into Equation 4:
\[ a = 5(1) = 5 \]

### Final Answer
Thus, the values of \( a \) and \( b \) for which the given pair of linear equations has an infinite number of solutions are:
\[ \boxed{a = 5, b = 1} \]