Question

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

x + 2y = 1, (a – b)x + (a + b)y = a + b – 2

Answer 1

The given pair of linear equations are:

x + 2y = 1   ......(i)

(a – b)x + (a + b)y = a + b – 2   ......(ii)

On comparing with ax + by = c = 0, we get

a₁ = 1, b₁ = 2, c₁ = – 1

a₂ = (a – b), b₂ = (a + b), c₂ = – (a + b – 2)

`a_1/a_2 = 1/(a - b)`

`b_1/b_2 = 2/(a + b)`

`c_1/c_2 = 1/(a + b - 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, `1/(a - b) = 2/(a + b) = 1/(a + b - 2)`

Taking first two parts,

`1/(a - b) = 2/(a + b)`

a + b = 2(a – b)

a = 3b   .......(iii)

Taking last two parts,

`2/(a + b) = 1/(a + b - 2)`

2(a + b – 2) = (a + b)

a + b = 4   .......(iv)

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

3b + b = 4

4b = 4

b = 1

Put the value of b in equation (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.