Questions & Answers

Question

Answers

A. Inconsistent

B. Consistent

C. Dependent

D. Empty

Answer
Verified

So, in this problem we need to find out the kind of linear equations that have no solution.

Let there be a system of two linear equations given as:

\[

{a_1}x + {b_1}y + {c_1} = 0 \\

{a_2}x + {b_2}y + {c_2} = 0 \\

\]

such that the lines represents by the equations \[{a_1}x + {b_1}y + {c_1} = 0\] and \[{a_2}x + {b_2}y + {c_2} = 0\] respectively are parallel lines which never intersect :

For a system of linear equations to have a unique solution, the lines should intersect. But since the lines in this case are parallel, it means they will never intersect anywhere and hence it will have no solution. Thus, such a system of linear equations is called inconsistent.

So we can say that an inconsistent pair of linear equations has no solution such that:

For a system of equations if $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$ then, the system of linear equations is inconsistent having no solution.

Hence an inconsistent system of linear equations has no solution.

So, our correct answer is option A.

\[

{a_1}x + {b_1}y + {c_1} = 0 \\

{a_2}x + {b_2}y + {c_2} = 0 \\

\]

Case 1) If, $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$ , then the system of equations is consistent having a unique solution, since the lines represented by their graphs intersect at one point.

Case 2) If $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$, then the system of equations is consistent having infinitely many solutions, since the lines represented by their graphs are coincident.