For what value of $k$, the following system of equations will represent the coincident lines?
$x+2y+7=0$
$2x+ky+14=0$

Given: 

The given system of equations is:

$x+2y+7=0$

$2x+ky+14=0$

To do: 

We have to find the value of $k$ for which the given system of equations represent the coincident lines.

Solution:

The given system of equations is,

$x+2y+7=0$

$2x+ky+14=0$

Two coincident lines will have infinite solutions.

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 represent the coincident lines 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,

$a_1=1, b_1=2, c_1=7$ and $a_2=2, b_2=k, c_2=14$

Therefore,

$\frac{1}{2}=\frac{2}{k}=\frac{7}{14}$

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

$k= 2\times2$

$k=4$

The value of $k$ for which the given system of equations represent the coincident lines is $4$.

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Updated on: 10-Oct-2022

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