For what value of k, the following pair of linear equations has infinitely many solutions?

10x + 5y – (k – 5) = 0

20x + 10y – k = 0

Given:

Equation 1: 10x + 5y = (k – 5) Equation 2: 20x + 10y = k

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 = 10

a₂ = 20

b₁ = 5

b₂ = 10

c₁ = k – 5

c₂ = k

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

…(ii)

On solving the equality (ii) we get

5k = 10( k – 5 ) ⇒ 5k = 10k – 50 ⇒ 5k = 50 ⇒ k = 10

The value of k for which the system of equations has infinitely many solution is k = 10