### Theory:

In the previous theory, we have learnt how to find the solutions using the cross multiplication method.

Let us discuss the consistency of the solutions of the linear equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) based on the solutions \(x=\frac{(b_1c_2-b_2c_1)}{(a_1b_2-a_2b_1)}\) and \(y=\frac{(c_2a_1-c_1a_2)}{(b_1a_2-b_2a_1)}\).

**: \(a_1b_2-a_2b_1\) \(\neq\) \(0\)**

*Case 1*\(\Rightarrow a_1b_2 \neq a_2b_1\)

\(\Rightarrow \frac{a_1}{a_2} \neq \frac{b_1}{b_2}\)

In this case, the system of linear equations has a unique solution.

**: \(a_1b_2-a_2b_1\) \(=\) \(0\)**

*Case 2*\(\Rightarrow a_1b_2 = a_2b_1\)

\(\Rightarrow \frac{a_1}{a_2} = \frac{b_1}{b_2}\)

Suppose \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = k\)(Constant)

Then, \(\frac{a_1}{a_2} = k\) and \(\frac{b_1}{b_2} = k\).

This implies, \(a_1 = ka_2\) and \(b_1 = kb_2\).

Substitute the above values in \(a_1x + b_1y + c_1 = 0\).

We have \(ka_2x + kb_2y + c_1 = 0\).

Thus, \(k(a_1x + b_1y) + c_1 = 0\).

Hence, the equations \(k(a_1x + b_1y) + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) can be satisfied only if \(c_1 = kc_2\).

That is when \(\frac{c_1}{c_2} = k\).

Therefore we say that if \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\) \(=\) \(k\)(constant), then the pair of linear equations have infinitely many solutions.

Also, if \(c_1 \neq kc_2\) both the linear equations cannot satisfy one and another.

In such a condition, the pair of linear equations will have no solution.

Now, let us summarise the condition for the system to be consistent and inconsistent.

**(1)**If \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\) then the system of equations has a unique solution. Hence, the given system of equations is consistent. In the graphical representation, if the equations are consistent, then the lines intersect at only one point.

**(2)**If \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\), then the system of equations has infinitely many number of solutions. Hence, the given system of equations is consistent. In the graphical representation, if the equations are consistent, then the lines coincide with each other.

**(3)**If \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\), then the system of equations has no solution. Hence, the given system of equations is inconsistent. In the graphical representation, if the equations are inconsistent, then the lines are parallel to each other.