#### Question

Find the values of a and b for which the following system of equations has infinitely many solutions:

2x - (2a + 5)y = 5

(2b + 1)x - 9y = 15

#### Solution

The given system of equations is

2x - (2a + 5)y - 5 = 0

(2b + 1)x - 9y - 15 = 0

It is of the form

`a_1x + b_1y + c_1 = 0` `

a_2x + b_2y + c_2 = 0`

Where `a_1 = x , b_1 = -(2a + 5), c_1 = -5`

And `a_2 = (2b = 1),b_2 = -9, c_2 = -15`

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

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

`=> 2/(2b + 1) = (-(2a + 5))/(-9) = (-5)/(-15)`

`=> 2/(2b + 1) = 1/3 and (2a + 5)/9 = 1/3`

`=> 6 = 2b + 1 and (3(2a + 5))/9 = 1`

`=> 6 - 1 = 2b and 2a + 5 = 3`

=> 5 = 2b and 2a = -2

`=> 5/2 = b and a = (-2)/2 = -1`

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

`if a = -1 and b = 5/2 `