Find the value of $k$ for which the following system of equations has no solution:
$x\ +\ 2y\ =\ 0$$2x\ +\ ky\ =\ 5$

Given: 

The given system of equations is:

$x\ +\ 2y\ =\ 0$

$2x\ +\ ky\ =\ 5$

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:

$x\ +\ 2y\ =0$

$2x\ +\ ky\ -\ 5=0$

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,

$a_1=1, b_1=2, c_1=0$ and $a_2=2, b_2=k, c_2=-5$

Therefore,

$\frac{1}{2}=\frac{2}{k}≠\frac{0}{-5}$

$\frac{1}{2}=\frac{2}{k}≠0$

$\frac{1}{2}=\frac{2}{k}$

$k\times1=2\times2$

$k=4$

The value of $k$ for which the given system of equations has no solution is $4$. 

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Updated on: 10-Oct-2022

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