Find the value of $k$ for which the following system of equations having infinitely many solutions:

$8x\ +\ 5y\ =\ 9$
$kx\ +\ 10y\ =\ 18$


Given: The given equation are  $8x + 5y= 9$;$kx + 10y =18$ 

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:

$8x + 5y= 9$

$kx + 10y =18$

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}} \ $


Here, $a_1 = 8, b_1=5, c_1=-9 \ and \ a_2=k, b_2=10, c_2=-18 $

$\frac{8}{k}  =\frac{5}{10} =\frac{-9}{-18} \ $

Now ,  $\frac{8}{k}  =\frac{5}{10}$

$\Rightarrow 8\times10 = 5k$

$\Rightarrow k = \frac{8\times10}{5}$

$\Rightarrow k = 16$

Hence, the  system of equations having infinitely many solutions if $k = 16$

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Updated on: 10-Oct-2022

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