Question

Obtain the condition for the following system of linear equations to have a unique solution.
$
  ax + by = c \\
  lx + my = n \\
 $

Answer 1

Hint: Here, we will proceed by comparing the given pair of linear equations with any general pair of linear equations i.e., ${a_1}x + {b_1}y + {c_1} = 0$ and ${a_2}x + {b_2}y + {c_2} = 0$. Then using the condition for having unique solution i.e., $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$.

Complete Step-by-Step solution:
The given system of linear equations is $
  ax + by = c \\
   \Rightarrow ax + by - c = 0{\text{ }} \to {\text{(1)}} \\
 $ and $
  lx + my = n \\
   \Rightarrow lx + my - n = 0{\text{ }} \to {\text{(2)}} \\
 $
As we know that for any pair of linear equations ${a_1}x + {b_1}y + {c_1} = 0{\text{ }} \to {\text{(3)}}$ and \[\] to have unique solution (consistent solution), the condition which must be satisfied is that the ratio of the coefficients of x should not be equal to the ratio of the coefficients of y in the pair of linear equations.
The condition is $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}{\text{ }} \to (5{\text{)}}$
By comparing equations (1) and (3), we get
${a_1} = a,{b_1} = b,{c_1} = - c$
By comparing equations (2) and (4), we get
${a_2} = l,{b_2} = m,{c_2} = - n$
For the given pair of linear equations to have a unique solution, equation (5) must be satisfied
By equation (5), we can write
$
  \dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}{\text{ }} \\
   \Rightarrow \dfrac{a}{l} \ne \dfrac{b}{m} \\
 $
Therefore, for the given linear system of equations to have unique solution, the necessary condition is that the ratio $\dfrac{a}{l}$ should not be equal to the ratio $\dfrac{b}{m}$ i.e., $\dfrac{a}{l} \ne \dfrac{b}{m}$.

Note: A pair of linear equations which are given by ${a_1}x + {b_1}y + {c_1} = 0$ and ${a_2}x + {b_2}y + {c_2} = 0$ can also have infinitely many solutions, the condition $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$ should always be satisfied. In this particular problem, the condition doesn’t depend on the constant terms for the pair of linear equations to have unique solutions.