Define Consistency

A pair of linear equations is consistent if it has at least one common solution (point of intersection).

$\text{Consistent} \implies \text{Solution Exists}$

Identify Types of Consistent Lines

Two types of lines provide consistent solutions: 1) Intersecting lines (unique solution) and 2) Coincident lines (infinitely many solutions).

$\frac{a_1}{a_2} \neq \frac{b_1}{b_2} \text{ (Intersecting) OR } \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \text{ (Coincident)}$

Conclusion

Since consistent systems cover both scenarios, the lines must be either intersecting or coincident.

$\text{Consistent} = \text{Intersecting} \cup \text{Coincident}$