Question 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
Compare the given equations with
a1x+b1y+c1=0 and a2x+b2y+c2=0.
a1a2=2a−b
b1b2=3a+b
c1c2=−7−(3a+b−2)=7(3a+b−2)
For infinitely many solutions, a1a2=b1b2=c1c2
2a−b=73a+b−2
⇒2(3a + b - 2) = 7 (a-b)
⇒6a + 2b - 4 = 7a - 7b
⇒a - 9b = -4 ... (i)
2a−b=3a+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.