Obtain the condition for the following system of linear equations to have a unique solution
$a x+b y=c$
$l x+m y=n$

Given: 

The given system of equations is:

$a x+b y=c$
$l x+m y=n$

To do: 

We have to find a condition for the given system of linear equations to have a unique solution.

Solution:

The given system of equations can be written as:

$a x+b y-c=0$
$l x+m y-n=0$

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 a unique solution is:

$\frac{a_{1}}{a_{2}} \ ≠ \frac{b_{1}}{b_{2}}$

Comparing the given system of equations with the standard form of equations, we have,

$a_1=a, b_1=b, c_1=-c$ and $a_2=l, b_2=m, c_2=-n$

Therefore,

$\frac{a}{l}≠\frac{b}{m}$

$a\times m≠ b\times l$

$am≠bl$

The condition for the given system of linear equations to have a unique solution is  $am≠bl$.

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Updated on: 10-Oct-2022

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