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

a_{1} = , b_{1} = 2, c_{1} = - 1

And a_{2} = (a - b), b_{2} = (a + b), c_{2} = - (a + b - 2)

a_{1} /a_{2} =

b_{1} /b_{2} =

c_{1} /c_{2} =

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

a_{1}/a_{2} = b_{1}/b_{2}c_{1}/c_{2}(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.

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