Question

Determine the values of a and b so that the following system of linear equations have infinitely many solutions:

(2a - 1)x + 3y - 5 = 0

3x + (b - 1)y - 2 = 0

Answer 1

The given system of equations may be written as

(2a - 1)x + 3y - 5 = 0

3x + (b - 1)y - 2 = 0

It is of the form

`a_1x + b_1y + c_1 = 0`

`a_2x + b_2y + c_2 = 0`

Where `a_1 = 2a, b_1 = 3, c_1 = -5`

And `a_2 = 3, b_2 = b - 1, c_2 = -2`

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

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

`=> (2a - 1)/3 = 3/(b - 1) = (-5)/(-2)`

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

`=> 2(2a - 1) = 5 xx 3 and 3 xx 2 = 5(b - 1)`

`=> 4a - 2 = 15 and  6 = 5b - 5`

`=> 4a = 15 + 2 and 6 + 5 = 5b`

`=> a = 17/4 and 11/5 = b`

`=> a = 17/4 and b = 11/5`

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

if `a = 17/4 and b = 11/5`