3.2 Conditions for Consistency

Standard form: $1x + 3y - 6 = 0$ and $2x - 3y - 12 = 0$.

Identify coefficients: $a_1=1, b_1=3, c_1=-6$; $a_2=2, b_2=-3, c_2=-12$.

Check ratios: $a_1/a_2 = 1/2$. $b_1/b_2 = 3/-3 = -1$.

Since $1/2 \neq -1$, i.e., $a_1/a_2 \neq b_1/b_2$.

The lines intersect at a unique point. The system is consistent.