Question

Find the value of $k$ for which the system of equations has a unique solution:
$x - ky = 2$, $3x + 2y = - 5$

Answer 1

There are certain predefined conditions to check whether a system of equations is unique, infinite or having no solution. To check whether the system of equations is having a unique solution we will justify the following expression.
$\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$
Where, ${a_1}$ and ${a_2}$ are the coefficient and ${b_1}$ and ${b_2}$ are the coefficient of $y$.
Also ${c_1}$ and ${c_2}$ are used to represent the constant term in the expression.
We will substitute the values in the above expression to find the value of $k$.

Complete step-by-step solution:
Given: $x - ky = 2$ , $3x + 2y = - 5$
For the given equations we will find the values of ${a_1}$ ,${b_1}$ ,${c_1}$ and ${a_2}$ ,${b_2}$ ,${c_2}$which can be expressed as:
${a_1} = 1$ ,${b_1} = - k$ ,${c_1} = - 2$ and ${a_2} = 3$ ,${b_2} = 2$ ,${c_2} = 5$
To check whether a system of equations has a unique solution or not, the following expression must be satisfied.
$\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$
We will substitute 1 for ${a_1}$ ,$k$ for ${b_1}$ , 2 for ${a_2}$ and 2 for ${b_2}$ in the above expression to find the value of $k$ .
$\dfrac{1}{3} \ne \dfrac{k}{2}$
We will rearrange the above expression to find the value of $k$ .
$k \ne \dfrac{2}{3}$
That means $k$ can have any value but it can’t be $\dfrac{2}{3}$ .
Hence for a given system of equations to have a unique solution, the value of $k$ must not be $\dfrac{2}{3}$ .

Note: The given system of equation has unique solution therefore the condition that must be satisfied is $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$ . If the system of equation has infinite solution, then following condition must be satisfied:
$\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$
For the system of equation which has no solution following condition must be satisfied:
$\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$

These conditions are predefined and can be used to check the solution of the system of equations. Also if it is already given in the question that the system has a unique, infinite or no solution, then we can use these predefined conditions to find the unknown variables.