Question

Find the values of p and q for which the following system of linear equations has infinite a number of solutions:

2x - 3y = 9

(p + q)x + (2p - q)y = 3(p + q + 1)

Answer 1

The given system of equations may be written as

2x - 3y - 9 = 0

(p + q)x + (2p - q)y = 3(p + q + 1) = 0

It is of the form

`a_1x + b_1y + c_1 = 0`

`a_2x + b_2y + c_2 = 0`

Where `a_1 = 2, b_1 = 3, c_1 = -9`

And `a_2 = p + 1, b_2 = 2p - q, c_2 = -3(p + q + 1)`

The given system of equations will have infinite number of solutions, if

`a_1/a_2 - b_1/b_2 = c_1/c_2`

`=> 2/(p + q) = 2/(2p - q) = (-9)/(-3(p + q + 1))`

`=> 2/(p + q) = 3/(2p - q) = 3/(p + q + 1)`

=> 2(2p - q) = 3(p + q) and p + q + 1 = 2p - q

`=> 4p - 2q = 3p + 3q and -2p + p + q + q = -1`

=> p = 5q = 0 and -p + 2q = -1

=> -3q = -1

`=> q = 1/3`

Puttign q = 1/3 in p - 5q we get

`p -5(1/3) = 0`

`=> p = 5/3`

Hence, the given system of equations will have infinitely many solutions,

if  p = 5/3 and q = 1/3