Question

For which values of p and q will the following pair of linear equations have infinitely many solutions: ( p – 1 )x + 3y = 2 , 6x + ( 2 – q )y = 6

Answer 1

Let (p-1) and (2-q) be 'm' and 'n' respectively.
Therefore, mx + 3y = 2 (eq. 1) ;  6x + ny = 6 (eq. 2)
The solutions can be infinite only if ny = (6 - 6x) and mx = (2 - 3y)
As ny = (6 - 6x) = 3(2 - 2x) and 3y = (2 - mx) ,  ny = 3(3y) = 9y and mx = 2x.
Therefore, n = 9  and m =  2
m = (p - 1) = 2 ,      p - 1 = 2
                             p = 2+1 = 3
n = (2 - q) = 9 ,       2 - q = 9
                             2 - 9 = q = -7