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

1.	Find the value of k for the system of equations having infinitely many solution:4x + 5y = 3;kx + 15y = 9
Answer» 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.

Find the value of k for the system of equations having infinitely many solution:4x + 5y = 3;kx + 15y = 9

Answer»

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.