Question

Explain Consistent and Inconsistent Systems and it's representation.

Answer 1

Consistent and Inconsistent Systems and it's representation:

A pair of linear equations in two variables in general can be represented as:

$\begin{array}{l}{a}_{1}x+b_{1}y+c_1\end{array}=0$

$\begin{array}{l}{a}_{2}x+b_{2}y+c_2\end{array}=0$

Consistent System: If both the lines intersect at a point, then there exists a unique solution to the pair of linear equations. In such a case, the pair of linear equations is said to be consistent.

Algebraically, if $\begin{array}{l}\frac{a_1}{a_2}\ne \frac{b_1}{b_2}\end{array}$then, the linear equations’ pair is consistent.

Let these lines coincide with each other, then there exist infinitely many solutions since a line consists of infinite points. In such a case, the pair of linear equations is said to be dependent and consistent. As represented in the graph below, the pair of lines coincides and therefore, dependent and consistent.

Algebraically, when $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

Inconsistent System: Let both the lines to be parallel to each other, then there exists no solution, because the lines never intersect.

Algebraically, when $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$