Given:

Equation 1: 3x – 2y = 0

Equation 2: kx + 5y = 0

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

………(i)

According to the problem:

a₁ = 3

a₂ = k

b₁ = – 2

b₂ = 5

c₁ = 0

c₂ = 0

In this problem since c₁ & c₂ = 0 so which is undefined.

So for this problem the system of linear equations will have infinite solutions if

.......(ii)

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

⇒ – 2k = 15

⇒ k =

The value of k for which the system of equations has infinitely many solution is k =