Write the value of k for which the system of equations has infinitely many solutions.
Question:

Write the value of k for which the system of equations has infinitely many solutions.

$2 x-y=5$

$6 x+k y=15$

Solution:

The given systems of equations are

$2 x-y=5$

$6 x+k y=15$

$a_{1}=2, a_{2}=6, b_{1}=1, b_{2}=k, c_{1}=5, c_{2}=15$

$\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}$

For the equations to have infinite number of solutions, $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$

$\frac{2}{6}=\frac{-1}{k}$

By cross Multiplication we get,

$2 k=-6$

$k=\frac{-6}{2}$

$k=-3$

Hence the value of $k$ is $-3$ when equations has infinitely many solutions.