Question

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

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

3x + (b - 2)y = 3

Answer 1

The given system of equations is

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

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

It is of the form

`a_1x + b_1y + c_1 = 0`

`a_2x + b_2y + c_2 = 0`

Where `a_1 = 2a - 1, b_1 = -3,c_1 = -5`

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

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 - 2) = (-5)/(-3)`

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

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

=> 2a = 5 + 1 and -9 + 10 = 5b

`=> a = 6/2 and 1 = 5b`

`=> a= 3 and 1/5 = b`

`=> a = 3 and b = 1/5`

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

if `a = 3 and b = 1/5`