The given equation is $2x + 3y - 8 = 0$, where $a_1 = 2$, $b_1 = 3$, $c_1 = -8$. For intersecting lines, the ratios of coefficients must be unequal ($\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$). For parallel lines, the ratios of x and y coefficients must be equal but different from the ratio of constants ($\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$). For coincident lines, all three ratios must be equal ($\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$).