Question

Give solution emidiately ​

Answer 1

$\huge\frak\red{Solution}$

$\bold{ \: The \: \: given \: \: system \: \: is:} \\ \\ \tt{3x + 5y = 0 - - - (i)} \\ \tt{kx + 10y = 0 - - - (ii)}$

$\textbf{This \: is \: a \: homogenous \: system \: of \: linear \: differential \: equations,} \\ \textbf{so \: it \: has \: a \: zero \: solution \: i.e \: x = y = 0}$

$\textbf{ \: But \: to \: have \: a \: non \: zero \: solution \: it \: must \: have \: infinite \: solutions} \\ \textbf{ \: for \: this \: we \: have \: ,} \\ \tt{ \frac{ a_{1}}{ a_{2} }} = \frac{ b_{1} }{ b_{2}} \\ \\ \tt{\implies \frac{3}{k} = \frac{5}{10} } \\ \\ \tt{\implies \: 30 = 5k} \\ \\ \tt{\implies \: k \: = 6} \\ \\ \huge\boxed{Therefore, \: k = 6}$