Question

Find the values of a and b for which the system of linear equations has an infinite number of solutions:
2x + 3y = 7, (a + b)x + (2a - b)y = 21.

Answer 1

The given system of equations can be written as
2x + 3y - 7 = 0                                               ….(i)
(a + b)x + (2a - b)y – 21 = 0                        ….(ii)
This system 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 = 2a - b, c_2= – 21`
For the given system of linear equations to have an infinite number of solutions, we must have:
`(a_1)/(a_2) = (b_1)/(b_2) = (c_1)/(c_2)`
`⇒2/(a+b) = 3/(2a−b) = (−7)/(−21)`
`⇒ 2/(a+b) = (−7)/(−21) = 1/3 and 3/(2a−b) = (−7)/(−21) = 1/3`
⇒ a + b = 6 and 2a –b = 9
Adding a + b = 6 and 2a – b = 9 ,we get
3a = 15 ⇒ a = `15/3` = 3
Now substituting a = 5 in a + b = 6, we have
5 + b = 6 ⇒ b = 6 – 5 = 1
Hence, a = 5 and b = 1.