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.