Consistency of Three Equations in Two Variables

4.3.2 Consistency of Three Equations in Two Variables

Consider three linear equations in the two variables $x,y$:

These are said to be consistent if they have a common solution.

Theorem. The necessary condition for the three equations to be consistent is

Proof (sketch). Solve the last two equations for $x,y$ using Cramer's Rule (treating them as a 2-variable system, provided $\begin{vmatrix}a_2&b_2\\a_3&b_3\end{vmatrix}\ne0$):

Substituting these into the first equation $a_1x+b_1y+c_1=0$ and clearing the common denominator $a_2b_3-a_3b_2$ produces, after regrouping the terms, exactly the cofactor expansion of $\begin{vmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{vmatrix}$ along row 1. So the substituted equation is equivalent to this determinant being $0$.

Important caveat. This condition is necessary but not sufficient — it can hold even when the equations have no common solution (e.g. three mutually parallel lines, which never meet, still make the determinant vanish because the rows become proportional). So after the determinant test passes, you must still check geometrically/algebraically that a genuine common solution exists.

Worked Examples

Example 1. Verify the consistency of $2x+2y=-2,\ x+y=-1,\ 3x+3y=-5$.

Step 1: $\begin{vmatrix}2&2&2\\1&1&1\\3&3&5\end{vmatrix}=2(5-3)-2(5-3)+2(3-3)=4-4+0=0$. …