Q.Examine the consistency of the following equations: $2x+3y-4=0,\ x+2y=3,\ 3x+4y+5=0$

Three linear equations $a_ix+b_iy+c_i=0$ ($i=1,2,3$) in the two unknowns $x,y$ are called consistent if they share a common solution — geometrically, if the three lines they represent all pass through one point. The determinant test is:

This is proved by solving the last two equations with Cramer's Rule and substituting into the first; the algebra collapses exactly into this determinant. Crucially, the condition is necessary but not sufficient — three mutually parallel lines (which never meet at all) also make this determinant vanish, because their coefficient rows are proportional. So after the determinant test passes, you still need to verify a genuine common …