Question

If the pair of linear equations 3x-5y=11 and (5a+b)x-(7a-b)y=4(3a-2b)+1 have infinitely many solutions then find the value of a and b

Answer 1

3x-5y-11-0
(5a+b)x-(7a-b)y-[4(3a-2b)+1) = 0

from the equation, we have
a1 =3, b1= -5 c1= -11
a2=(5a+b), b2= -(7a-b), c2= -(12a-8b+1)

as the equations have infinitely many solution,

so, a1/a2 = b1/b2 = c1/c2
taking a1/a2=b1/b2,
we have 3/(5a+b) = -5/-(7a-b)
or 5(5a+b) = 3(7a-b)
or 25a-21a+5b+3b = 0
or 4a+8b=0
or a+2b=0 .....eq1)

taking b1/b2=c1/c2
-5/-(7a-b) = -11/-(12a-8b+1)
or 77a-11b = 60a-40b+5
or 17a+29b = 5 .......2)

on solving from 1) & 2)
we get b= -1 a=2 Ans.