Find the value of $k$ for which the system of equations $2 x+3 y-5=0$ and $4 x+k y-10=0$ has infinite number of solutions.

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Find the value of $k$ for which the system of equations $2 x+3 y-5=0$ and $4 x+k y-10=0$ has infinite number of solutions.

Find the value of $k$ for which the system of equations $2 x+3 y-5=0$ and $4 x+k y-10=0$ has infinite number of solutions.

Solution:

The given system of equations is
$\begin{array}{ll} 2 \mathrm{x}+3 \mathrm{y}-5=0\dots \dots(i) \\ 4 \mathrm{x}+\mathrm{ky}-10=0 \ldots \ldots(\mathrm{ii}) \\ \end{array}$
Here, $a_{1}=2, b_{1}=3, c_{1}=-5, a_{2}=4, b_{2}=k$ and $c_{2}=-10$
For the given system of equations, to have an infinite number of solutions, we must have
$\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$ $\Rightarrow \frac{2}{4}=\frac{3}{k}=\frac{-5}{-10}$
$\Rightarrow \frac{1}{2}=\frac{3}{k}=\frac{1}{2}$
$\Rightarrow \mathrm{k}=6$
As a result, $\mathrm{k}=6$