To check consistency from coefficient ratios alone (without comparing constant terms), the equations must satisfy —

• a$\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2}$ or both constants are zero
• b$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2}$ always
• c$a_1 = a_2$ and $b_1 = b_2$
• d$c_1 = c_2 \ne 0$