Question

The pair of equations \[x + 2y + 5 = 0\] and \[-{\text{ }}3x-{\text{6}}y + 1 = 0\] has
(A) A unique solution
(B) Exactly two solutions
(C) Infinitely many solutions
(D) No solution

Answer 1

Hint: We have been given a pair of equations, there are certain conditions of a pair of equations to have unique, finite, infinite and no solutions. Knowing these conditions and comparing the same to the given equations, we can easily make conclusion about the existence of the solution of the given pair of equations

Complete step by step solution:
If the two pair of equations are:
\[
  {a_1}x + {b_1}y + {c_1} = 0 \\
  {a_2}x + {b_2}y + {c_2} = 0 \;
 \]
Then the solutions varies as:
If $ \dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}} $ , then then the solution is unique, if $ \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}} $ , then there and infinitely many solutions and if $ \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}} $ , then the solution does not exist.
The given equations are:
\[x + 2y + 5 = 0\]
\[-{\text{ }}3x-{\text{6}}y + 1 = 0\]
Here, the respective coefficients of variables are:
 $ {a_1} = 1 $, $ {b_1} = 2 $ , $ {c_1} = 5 $
 $ {a_2} = - 3 $, $ {b_2} = - 6 $, $ {c_2} = 1 $
Now, dividing the coefficient of x, y and z of respective equations, we get:
\[
\Rightarrow \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{1}{{ - 3}} \\
\Rightarrow \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{2}{{ - 6}} \Rightarrow \dfrac{1}{{ - 3}} \\
\Rightarrow \dfrac{{{c_1}}}{{{c_2}}} = \dfrac{5}{1} \;
 \]
If taken together their relationship is given as:
 $
\Rightarrow \dfrac{1}{{ - 3}} = \dfrac{1}{{ - 3}} \ne \dfrac{5}{1} \\
\Rightarrow \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}} \;
  $
We know that if $ \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}} $ , then the solution does not exist. Therefore, the given equations have no solution.
So, the correct answer is “Option D”.

Note: The equations whose solutions exist are known as consistent and when the solution does not exist, the pair is known as inconsistent.
So, the given pair of equations was inconsistent.
The given equations were linear, representing a line. If there is no solution to the equations, this shows that the lines never intersects. When the lines never intersect, they are known as parallel lines. So the lines satisfying the condition $ \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}} $ are parallel