Question

Find the value of $k$ for which the given system of equations has a unique solution.$\left( {k - 3} \right)x + 3y = k;kx + ky = 12$
A.$k \ne 3$
B.$k \ne 9$
C.$k \ne 6$
D.$k \ne 2$

Answer 1

Hint: Here, we have to find the value of the variable for which the system of equations has a unique solution. First, we will equate the given system of equations with the general system of equations. Then we have to use the condition for the system of equations which has a unique solution. Thus we have to find the value.

Complete step-by-step answer:
We are given the system of linear equations.
The general form of a linear equation is given by $ax + by + c = 0$.
Comparing the given system of linear equation with the general system of linear equations, we get
\[{a_1} = \left( {k - 3} \right) \\
  {a_2} = k \\
  {b_1} = 3 \\
  {b_2} = k \\
  {c_1} = - k \\
  {c_2} = - 12 \\ \]
Now we know that condition for which the given system has a unique solution is given by $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$.
Since the given system of equation have unique solution, we have
$\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$
Substituting \[{a_1} = \left( {k - 3} \right),{a_2} = k,{b_1} = 3\] and \[{b_2} = k\] in $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$we get
$ \Rightarrow \dfrac{{(k - 3)}}{k} \ne \dfrac{3}{k}$
By cross multiplication, we have
$ \Rightarrow k(k - 3) \ne 3k$
 Multiplying the terms, we have
$ \Rightarrow {k^2} - 3k \ne 3k$
 Adding the terms, we have
$ \Rightarrow {k^2} - 3k - 3k \ne 0$
 $ \Rightarrow {k^2} - 6k \ne 0$
 Adding $6k$ on both the side, we get
$ \Rightarrow {k^2} \ne 6k$
Dividing both sides by $k$ the term, we get
$ \Rightarrow \dfrac{{{k^2}}}{k} \ne 6$
$ \Rightarrow k \ne 6$
Therefore, the system of linear equations has a unique solution if $k \ne 6$ .

Note: We should know the condition for which a system has a unique solution. A system of linear equations will have a unique solution if the two lines intersect at a point. i.e., if the two lines are neither parallel nor coincident. Essentially, the slopes of the two lines should be different. If the system does not have a unique solution, then their slopes must be equal.