Question

Which of the following pairs of equations represent an inconsistent system?
A.3x – 2y = 8
    2x + 3y = 1
B.3x – y = - 8
   3x – y = 24
C.x – y = m
   x + my = 1
D.5x – y = 10
   10x – 6y = 20

Answer 1

Hint: The question which are based on the linear equation is answered using the graphical representation of the given equations and observe the relation between the graph formed by both the equations and find which type of solution is shown by the pair of linear equations such as unique solution, no solutions, infinitely many solutions use these concepts of relation between the linear equations to approach to the solution of the given problem.

Complete step-by-step answer:
As we know that the general representation of couple of linear equations is shown as ${a_1}x + {b_1}y + {c_1} = 0$ and ${a_2}x + {b_2}y + {c_2} = 0$
So according to the system of linear equations there exists two types of system one is consistent system and another is inconsistent system of linear equations
The system is said to be consistent when for both the linear equations there exists at least one common set of values whereas in inconsistent system of linear equation there doesn’t exists a common set of values that satisfies the both equations
For inconsistent system there is a condition that would satisfied by the pair of linear equations which says that when the two lines that are graphical representation of pair of linear equations are parallel to each other due to no possible set of common values can exists this is called the no solution
The condition for the inconsistent system is $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$ here \[{a_1},{b_1},{c_1}\] and \[{a_2},{b_2},{c_2}\] are the real numbers of the pair of linear equations
So let’s check whether the pair of linear equations given in options satisfies the condition of inconsistent system or not
Check the pair of linear equations of 1st option
The given linear equations are 3x – 2y = 8 and 2x + 3y = 1
Comparing the given linear equations with the general representation of linear equations we get \[{a_1} = 3,{b_1} = - 2,{c_1} = - 8\] and \[{a_2} = 2,{b_2} = 3,{c_2} = - 1\]
Now substituting the given values in formula $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$
\[\dfrac{3}{{ - 2}} = \dfrac{{ - 2}}{3} \ne \dfrac{{ - 8}}{{ - 1}}\] as we can see that here $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$
Therefore this pair of linear equations doesn’t satisfies the condition of inconsistent system
Now for option 2nd the given pair of linear equations are 3x – y = - 8 and 3x – y = 24
Comparing the given linear equations with the general representation of linear equations we get \[{a_1} = 3,{b_1} = - 1,{c_1} = 8\] and \[{a_2} = 3,{b_2} = - 1,{c_2} = - 24\]
Now substituting the given values in formula $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$
\[\dfrac{3}{3} = \dfrac{{ - 1}}{{ - 1}} \ne \dfrac{{ - 8}}{{ - 24}}\] = \[\dfrac{1}{1} = \dfrac{1}{1} \ne \dfrac{{ - 8}}{{ - 24}}\]as we can see that here this pair of linear equations satisfies the condition of inconsistent system
Therefore the pair of linear equations are 3x – y = - 8 and 3x – y = 24 which represent the inconsistent system
For the pair of linear equations given in 3rd option i.e. x – y = m and x + my = 1
Comparing the given linear equations with the general representation of linear equations we get \[{a_1} = 1,{b_1} = - 1,{c_1} = - m\] and \[{a_2} = 1,{b_2} = m,{c_2} = - 1\]
Now substituting the given values in formula $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$
\[\dfrac{1}{1} = \dfrac{{ - 1}}{m} \ne \dfrac{{ - m}}{{ - 1}}\] as we can see that here $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$
Therefore this pair of linear equations doesn’t satisfies the condition of inconsistent system
The pair of linear equations in option 4th are 5x – y = 10 and 10x – 6y = 20
Comparing the given linear equations with the general representation of linear equations we get \[{a_1} = 5,{b_1} = - 1,{c_1} = - 10\] and \[{a_2} = 10,{b_2} = - 6,{c_2} = - 20\]
Now substituting the given values in formula $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$
\[\dfrac{5}{{10}} = \dfrac{{ - 1}}{{ - 6}} \ne \dfrac{{ - 10}}{{ - 20}}\] = \[\dfrac{1}{2} = \dfrac{{ - 1}}{{ - 6}} \ne \dfrac{1}{2}\]as we can see that here $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{c_1}}}{{{c_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$
Therefore this pair of linear equations doesn’t satisfies the condition of inconsistent system
Since the only pair given in option 2nd was satisfying the condition of an inconsistent system of linear equations, therefore option B is the correct option.

Note: In consistent system that we mentioned above the pairs of linear equations that represent the consistent systems follows the conditions such as when the two lines which represent the pair of linear equations intersect each other there exists a unique solutions which means there exists a set of pair of values that seems common to both the pair of linear equations the condition of this statement is followed by pair of linear equations when \[\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}\] also when the two lines are coincide together with each other there exists infinite common sets of values then this solution is called infinitely many solutions the pair of linear equations satisfies the above statement when $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$.