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Find the value of $k$ for which the following system of equations has no solution:
$2x\ +\ ky\ =\ 11$
$5x\ -\ 7y\ =\ 5$
Given:
The given system of equations is:
$2x\ +\ ky\ =\ 11$
$5x\ -\ 7y\ =\ 5$
To do:
We have to find the value of $k$ for which the given system of equations has no solution.
Solution:
The given system of equations can be written as:
$2x\ +\ ky\ -\ 11=0$
$5x\ -\ 7y\ -\ 5=0$
The standard form of system of equations of two variables is $a_{1} x+b_{1} y+c_{1}=0$ and $a_{2} x+b_{2} y-c_{2}=0$.
The condition for which the above system of equations has no solution is
$\frac{a_{1}}{a_{2}} \ =\frac{b_{1}}{b_{2}} ≠ \frac{c_{1}}{c_{2}} \ $
Comparing the given system of equations with the standard form of equations, we have,
$a_1=2, b_1=k, c_1=-11$ and $a_2=5, b_2=-7, c_2=-5$
Therefore,
$\frac{2}{5}=\frac{k}{-7}≠\frac{-11}{-5}$
$\frac{2}{5}=\frac{k}{-7}≠\frac{11}{5}$
$\frac{2}{5}=\frac{k}{-7}$
$-7\times2=k\times5$
$5k=-14$
$k=\frac{-14}{5}$
The value of $k$ for which the given system of equations has no solution is $\frac{-14}{5}$.