Question

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

4x + 5y = 3;

kx + 15y = 9

Answer 1

The given system of equations is: 

4x + 5y – 3 = 0 

kx + 15y – 9 = 0

The above equations are of the form 

a₁ x + b₁ y − c₁ = 0 

a₂ x + b₂ y − c₂ = 0 

Here, a₁ = 4, b₁ = 5, c₁ = -3 

a₂ = k, b₂ = 15, c₂ = -9 

So according to the question, 

For unique solution, the condition is 

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

\(\frac{4}{k} = \frac{5}{15} = \frac{-3}{-9}\)

\(\frac{4}{k} = \frac{1}{3}\)

⇒ k = 12 

Hence, the given system of equations will have infinitely many solutions, if k = 12.