Question

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

3x + 4y = 12

(a + b)x + 2(a - b)y = 5a - 1

Answer 1

The given system of equations is

3x + 4y - 12 = 0

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

It is of the form

`a_1x + b_1y + c_1 = 0` `

a_2x + b_2y + c_2 = 0`

Where `a_1 = 3, b_1 = 4, c_1 = -12`

And `a_2 = a + b, b_2 = 2(a - b), c_2 = - (5a - 1)`

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`

`=> 3/(a + b) = 4/(2(a - b)) = 12/(5a - 1)`

`=> 3/(a + b) = 2/(a - b) and  2/(a - b) = 12/(5a - 1)` 

`=> 3(a - b) = 2(a + b) and 2(5a - 1) = 12(a - b)`

`=> 3a - 3b = 2a + 2b  and 10a - 2 - 12a - 12b`

`=> 3a - 2a = 2b + 3b and 10a - 12a = -12b + 2`

`=> a = 5b and -2a = -12b + 2`

Substituting a = 5b in -2a = -12b + 2 we get

-2(5b) = -12b + 2

=> -10b = -12b + 2

=> 12b - 10b = 2

=> 2b = 2

=> b = 1

Putting b = 1 in a = 5b we get

`a = 5 xx 1 = 5`

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

if a = 5 and b = 1