Question

Find the value of k for which the system 3x + 5y = 0 and kx + 10y = 0 has a nonzero solution.

Answer 1

The given system is
$$\begin{gathered} 3x+5y=0.....\left(i\right) \\ kx+10y=0.....\left(ii\right) \end{gathered}$$
This is a homogeneous system of linear differential equation, so it always has a zero
solution i.e., x = y = 0.
But to have a nonzero solution, it must have infinitely many solutions.
For this, we have
$$\begin{gathered} \frac{a_1}{a_2}=\frac{b_1}{b_2} \\ \Rightarrow \frac{3}{k}=\frac{5}{10}=\frac{1}{2} \\ \Rightarrow k=6 \end{gathered}$$
Hence, k = 6.