Question

Write the set of values of a and b for which the following system of equations has infinitely many solutions.

2x + 3y = 7

2ax + (a + b) y = 28

Answer 1

Given:

Equation 1: 2x + 3y = 7

Equation 2: 2ax + (a + b)y = 28

Both the equations are in the form of :

a₁x + b₁y = c₁ & a₂x + b₂y = c₂ where

a₁ & a₂ are the coefficients of x

b₁ & b₂ are the coefficients of y

c₁ & c₂ are the constants

For the system of linear equations to have infinitely many solutions we must have

According to the problem:

a₁ = 2

a₂ = 2a

b₁ = 3

b₂ = (a + b)

c₁ = 7

c₂ = 28

Putting the above values in equation (i) we get:

To obtain the value of a & b we need to solve the above equality. First we solve the extreme left and extreme right of the equality to obtain the value of a.

⇒ 

⇒ 2a x 7 = 2 x 28

⇒ 14a = 56

⇒ a = 4

After obtaining the value of a we again solve the extreme left and middle portion of the equality (ii)

⇒ 2 x (4 + b) = 3 x 2 x 4

⇒ b + 4 = 12

⇒ b = 8

The value of a & b for which the system of equations has infinitely many solution is a = 4 & b = 8