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)y = 3a + b - 2

Answer 1

2x + 3y -7 = 0

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

Here `a_1 = 2, b_1 = 3,c_1 = -7`

`a_2 = (a - b), b_2 = (a + b), c_2 = -(3a + b - 2)`

`a_1/a_2 = 2/(a - b), b_1/b_2 = 3/(a + b), c_1/c_2 = (-7)/(-(3a + b - 2)) = (-7)/(3a + b - 2)`

For the equation to have infinitely many solutions, we have:

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

`2/(a - b) = 7/(3a + b -2)`

6a + 2b - 4 = 7a - 7b

a- 9b = -4 ......(1)

`2/(a -b) = 3/(a + b)`

a - 5b = 0  .....(2)

Subtracting (1) from (2), we obtain

4b = 4

b = 1

Substituting the value of b in equation (2), we obtain

a - 5 x  1 = 0

a = 5

Thus, the values of a and b are 5 and 1 respectively.