Question

Find the value of k for system of equations has a unique solution:

x + 2y = 3;

5x + ky + 7 = 0

Answer 1

The given system of equations is: 

x + 2y – 3 = 0

5x + ky + 7 = 0

The above equations are of the form 

a₁ x + b₁ y − c₁ = 0 

a₂ x + b₂ y − c₂ = 0 

Here, a₁ = 1, b₁ = 2, c₁ = −3 

a₂ = 5, b₂ = k, c₂ = 7 

So according to the question, 

For unique solution, the condition is 

\(\frac{a_1 }{a_2}\) ≠ \(\frac{b_1}{b_2}\) 

\(\frac{1}{5} ≠ \frac{2}{k}\)

⇒ k ≠ 10 

Hence, the given system of equations will have unique solution for all real values of k other than 10.