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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 $k = 16$
Updated on: 10-Oct-2022
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