For which values of a and b will the following pair of linear equations has infinitely many solutions?

x + 2y = 1 and (a - b)x + (a + b)y = a + b – 2

The given pair of linear equations are:

x + 2y = 1 …(i)

and (a-b)x + (a + b)y = a + b - 2 …(ii)

On comparing withax + by = c = 0 we get

a₁ = , b₁ = 2, c₁ = - 1

And a₂ = (a - b), b₂ = (a + b), c₂ = - (a + b - 2)

a₁ /a₂ =

b₁ /b₂ =

c₁ /c₂ =

For infinitely many solutions of the, pair of linear equations,

a₁/a₂ = b₁/b₂c₁/c₂(coincident lines)

so, = =

Taking first two parts,

=

a + b = 2(a - b)

a = 3b …(iii)

Taking last two parts,

=

2(a + b - 2) = (a + b)

a + b = 4 …(iv)

Now, put the value of a from Eq. (iii) in Eq. (iv), we get

3b + b = 4

4b = 4

b = 1

Put the value of b in Eq. (iii), we get

a = 3

So, the values (a,b) = (3,1) satisfies all the parts. Hence, required values of a and b are 3 and 1 respectively for which the given pair of linear equations has infinitely many solutions.