2. (i) For which values of a and b does the following pair of linear equations have an infinite number of solutions?
2x + 3y =7
(a – b)x + (a + b)y = 3a +b –2
2x + 3y -7 = 0
(a – b)x + (a + b)y - (3a +b –2) = 0
a1/a2 = 2/a-b = 1/2
b1/b2 = -7/a+b and
c1/c2 = -7/-(3a+b-2) = 7/(3a+b-2)
For infinitely many solutions,a1/a2 = b1/b2 = c1/c2
2/a-b = 7/3a+b-26a + 2b - 4 = 7a - 7b
a - 9b = -4 ... (i)
2/a-b = 3/a+b
2a + 2b = 3a - 3b
a - 5b = 0 ... (ii)
Subtracting equation (i) from (ii), we get
4b = 4
b = 1
Putting this value in equation (ii), we get
a - 5 × 1 = 0
a = 5
Hence, a = 5 and b = 1 are the values for which the given equations give infinitely many solutions.
