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 $\frac{-14}{5}$. 

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Updated on: 10-Oct-2022

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