Question

If the system of equations

2x + 3y = 7

2ax + (a + b)y = 28

has infinitely many solutions, then

A. a = 2b

B. b = 2a

C. a + 2b = 0

D. 2a + b = 0

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) and solving the extreme left and extreme right portion of the equality we get the value of a

⇒ 14a = 56

⇒ a = 4

We now put the value of a and solve for b

⇒ a + b = 12

⇒ b = 8

So

b = 2a

The correct relationship between a & b for which the system of equations has infinitely many solution is b = 2a