Theory:

In the previous topic, we have learnt how to find the solutions using the graphical method.
 
Now, let us see how to find the solution to the system of linear equations using the conditions.
 
Let us consider the system of linear equations in two variables.
 
\(a_{1}x + b_{1}y + c_{1} = 0\)
 
\(a_{2}x + b_{2}y + c_{2} = 0\)
 
Here, \(a_1, a_2, b_1, b_2, c_1, c_2\) are real numbers.
 
Now, let us see 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 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 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 graphical representation, if the equations are inconsistent, then the lines are parallel to each other.