Question

Find the value of k for the system of equations having infinitely many solution: 

2x + 3y – 5 = 0

6x + ky – 15 = 0

Answer 1

The given system of equations is: 

2x + 3y – 5 = 0

6x + ky – 15 = 0

The above equations are of the form 

a₁ x + b₁ y − c₁ = 0 

a₂ x + b₂ y − c₂ = 0 

Here, a₁ = 2, b₁ = 3, c₁ = -5 

a₂ = 6, b₂ = k, c₂ = -15 

So according to the question, 

For unique solution, the condition is 

\(\frac{a_1}{a_2}\) = \(\frac{b_1}{b_2}\) = \(\frac{c_1}{c_2}\) 

\(\frac{2}{6} = \frac{3}{k}\) 

⇒ k = 9 

Hence, the given system of equations will have infinitely many solutions, if k = 9.