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.
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