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Find the value of $k$ for which the following system of equations having infinitely many solution:
$4x\ +\ 5y\ =\ 3$
$kx\ +\ 15y\ =\ 9$
To do: Find the value of $k$ for which the following system of equations having infinitely many solutions.
Solution:
The given system of equation is:
$4x\ +\ 5y\ =\ 3$
$kx\ +\ 15y\ =\ 9$
The system of equation is of the form $a_{1} x+b_{1} y=c_{1}\ and\ a_{2} x+b_{2} y=c_{2}$
For the infinitely many solutions there is a condition
$\frac{a_{1}}{a_{2}} \ =\frac{b_{1}}{b_{2}} =\frac{c_{1}}{c_{2}} \ $
$\frac{4}{k} =\frac{5}{15} =\frac{3}{9} \ $
Now , $\frac{4}{k} =\frac{5}{15}$
$\Rightarrow 4\times15 = 5k$
$\Rightarrow k = \frac{4\times15}{5}$
$\Rightarrow k = 4\times3$
$\Rightarrow k = 12$
Hence, the system of equations having infinitely many solutions if $k = 12$
Updated on: 10-Oct-2022
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