ax + by = c and dx + ey = f

\( \Large \frac{a_{1}}{a_{2}}=\frac{a}{d}, \frac{b_{1}}{b_{2}}=\frac{b}{e}, \frac{c_{1}}{c_{2}}=\frac{c}{f} \)

Because, \( \Large \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \)

Therefore, \( \Large \frac{b}{e} = \frac{c}{f} \)

It represents a pair of parallel lines.

Because, \( \Large \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \)

Therefore, \( \Large \frac{a}{d} = \frac{b}{e}\)

Therefore, system has unique solution and represents a pair of intersecting lines.