Find the value of $k$ for which the system of equations$.2x+3y=5;\ 4x+ky=10$ has an infinite number of solutions.

Given: The system of equations $2x+3y=5;\ 4x+ky=10$ has an infinite number of solutions.

To do: To find the value of $k$.

Solution: 

Given equations are:

$2x+3y=5\ ....\ ( i)$

$4x+ky=10\ ...\ ( ii)$

Here, $a_1=2,\ b_1=3,\ c_1=5$

$a_2=4,\ b_2=k,\ c_2=10$

For infinitely many solution:

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

$\Rightarrow \frac{2}{4}=\frac{3}{k}=\frac{5}{10}$

$\Rightarrow \frac{3}{k}=\frac{1}{2}$

$\Rightarrow k=6$

Thus, for $k=6$, the given system of equations have infinitely many solutions.

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Updated on: 10-Oct-2022

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