Question

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

2x – 3y = 7

(a + b)x – (a + b – 3)y = 4a + b

Answer 1

The given system of equations may be written as

2x – 3y – 7 = 0

(a + b)x – (a + b – 3)y – 4a + b = 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 + b, b_2 = -(a + b - 3), c_2 = -(4a + b) `

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/(a + b) = (-3)/(-(a + b - 3)) = (-7)/(-(4a + b))`

`=> 2(a + b -3) = 3(a + b) and  3(4a + b) = 7(a + b - 3)`

`=> -6 = 3a - 2a + 3b - 2b  and 12a - 7a + 3b - 7b = 21`

`=> -6 = a + b and 5a - 4b = -21`

Now

a + b = –6

`=> a = -6 - b`

Substituting the value of a in 5a – 4b = –2 we get

5(–b – 6) – 4b = –21

`=> -5b - 30 - 4b = -21`

`=> -9b = -21 + 30`

`=> -9b = 9`

`=> b = 9/(-9) = -1`

Putting b = –1 in a = –b – 6 we get

a = –(–1) – 6 = 1 – 6 = –5

Hence, the given system of equations will have infinitely many solutions, if a = –5 and b = –1.