Question

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

2x + 3y = 7

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

Answer 1

The given system of equations is

2x + 3y - 7 = 0

(a - 1)x + (a + 2)y - 3a = 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 = -7`

And `a_2 = a - 1, b_2 = a + 1, c_2 = -3a`

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/(a - b) = 3/(a + 1) = (-7)/(-3a)`

`=> 2/(a - 1) = 3/(a + 2) = 7/(3a)`

`=> 2/(a - 1) = 3/(a + 2) and 3/(a + 2) = 7/(3a)`

=> 2(a + 2) = 3(a - 1) and 3 x 3a = 7(a + 2)

=> 2a - 4a = -3 and 9a = 7a + 14

=> -a = -7 and 2a = 14

`=> a = 7 and a = 14/2 = 7`

=> a = 7

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

if a = 7