Given:

Equation 1: 3x + 5y = 0

Equation 2: kx + 10y = 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

According to the problem:

a₁ = 3

a₂ = k

b₁ = 5

b₂ = 10

c₁ = 0

c₂ = 0

The equation will have a non zero solution only when it will satisfy a non trivial solution i.e. the equations should satisfy with values other than x = 0 & y = 0 .For the given system of equations the equation will have a non zero solution if

.......(i)

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

⇒ 5k = 30 ⇒ k = 6

The value of k for which the system of equations has non zero solution is k = 6