Question

For which value(s) of k will the pair of equations have no solution?
$kx+3y=k-3$;$12x+ky=k$; $k\ne 0$

Answer 1

Hint: First of all, we will compare both the equations with $ax+by+c=0$ and find the values of a, b and c with respect to the equations. After finding the values we will use the expression $\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}\ne \dfrac{{{c}_{1}}}{{{c}_{2}}}$, as it is mentioned that equations have no solution. Thus, we will find the value of k.
Complete step-by-step answer:
Now, as it is given in the question that we have to find the value of k for which the given equations have no solution, for that we will compare both equations with general equation i.e.
 ${{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0$ …………………(i)
${{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0$ …………………..(ii)
Now, equations are as follows:
$kx+3y=k-3$ ……………………(iii)
$12x+ky=k$ ……………………….(iv)
Comparing, the equation (i) with (iii) and (ii) with (iv) we will get,
${{a}_{1}}=k$, ${{b}_{1}}=3$and ${{c}_{1}}=-\left( k-3 \right)$
${{a}_{2}}=12$, ${{b}_{2}}=k$ and ${{c}_{2}}=-k$
Now, we will substitute this value in expression,
$\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}\ne \dfrac{{{c}_{1}}}{{{c}_{2}}}\Rightarrow \dfrac{k}{12}=\dfrac{3}{k}\ne \dfrac{-\left( k-3 \right)}{-k}$
Now, further simplifying the equation we will get,
$\dfrac{k}{12}=\dfrac{3}{k}$ and $\dfrac{3}{k}\ne \dfrac{-\left( k-3 \right)}{-k}$
$\Rightarrow {{k}^{2}}=36$ and $-3k\ne -{{k}^{2}}+3k$
$\Rightarrow k=\pm 6$ and $k\ne 6$
Now, from above derived values of k we have two different values $6$ and $-6$, but in the equation itself it can be observed that $k\ne 6$. Hence, the value of k is $-6$.
Thus, it can be said that if the value of k is $-6$, then the pair of equations have no solution.

Note: While solving such type of question students must compare the equations carefully and denote them with care too because if value of ${{a}_{1}}$ gets replaced with ${{a}_{2}}$ or vice-versa then whole problem will go wrong and required answer will not be obtained. Subject: Mathematics