Question

For what values of ‘\[m\]’, the pair of equations \[3x + my = 10\] and \[9x + 12y = 30\] have a unique solution?
A. \[4\]
B. Cannot be determined
C. Doesn’t exist
D. None of these

Answer 1

Hint:We are going to use the most eccentric concept of solving the simultaneous equations. So, solving (that is, multiplication of one of the equations with respect to the other equation, so as to eliminate the one term by means of addition, subtracting) the given equation to find the respective value of unknown ‘m’. As a result, substituting the values from the two equations respectively in the equation/condition \[\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}\], to satisfy/find the unique solution.

Complete step by step answer:
Since, here we have given the two simultaneous equations that is
\[ \Rightarrow 3x + my = 10\] And, … (i)
\[ \Rightarrow 9x + 12y = 30\] .… (ii)
As a result, to find the value of unknown(s) that is constant present in the given equations
So, first of all multiplying the equation (i) by \[3\], so as to subtract the equations to find out the unknown that is ‘\[m\]’, we get
\[9x + 3my = 30\] … (iii)

Now,solving these equations mathematically that is subtracting the equations (ii) and (iii) respectively, we get
\[9x + 12y = 30\]
\[ 9x + 3my = 30\]
\[- {\text{ }} - {\text{ }} = - \]
\[ \Rightarrow 12 - 3m = 0\]

Hence, the equation becomes
\[ \Rightarrow 12 - 3m = 0\]
\[ \Rightarrow 12 = 3m\]
The value of unknown ‘m’ is,
\[ \Rightarrow m = 4\]
Substituting the value of \[m = 4\] in equation (i), we get
\[ \Rightarrow 3x + 4y = 10\] … (iv)

Now, from (iv) and (ii) respectively, it seems that
\[ \Rightarrow 3x + 4y = 10\], and
\[ \Rightarrow 9x + 12y = 30\]
Where,
\[{a_1} = 3\], \[{a_2} = 9\], \[{b_1} = 4\], \[{b_2} = 12\], \[{c_1} = 10\] and \[{c_2} = 30\] respectively.
Hence, considering ratio that is
\[ \Rightarrow \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{3}{9} = \dfrac{1}{3}\] … (v)

Similarly,
\[ \Rightarrow \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{4}{{12}} = \dfrac{1}{3}\] … (vi)
And,
\[ \Rightarrow \dfrac{{{c_1}}}{{{c_2}}} = \dfrac{{10}}{{30}} = \dfrac{1}{3}\] … (vii)
From (v), (vi) and (vii),
It exists \[\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}\], which in term seems that the condition of parallel condition i.e. the given equations (representing the line) are parallel to each other.Hence, from the above solutions/calculations for ‘m’ equal to \[4\]; the pair of given equations has an unique solution as a parallel line.

Therefore, option A is correct.

Note:While adding or subtracting the certain equations happens only when the certain variables are same/equal such as \[11x\] and \[5x\], \[6y\] and \[9y\], etc. Remember the certain conditions such as for parallel lines, perpendicular lines, etc. represented by the conditions that is \[\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}\], \[{m_1} = {m_2}\], \[{m_1} = - \dfrac{1}{{{m_2}}}\], etc. (where, here ‘m’ is slope of the given lines used in this equation), so as to be sure of our final answer.