1. The problem asks: When is a system of equations inconsistent? 2. A system of equations is inconsistent if there is no solution that satisfies all equations simultaneously. 3. This happens when the equations represent parallel lines (in 2D) or planes that do not intersect (in 3D). 4. Mathematically, for a system \(Ax = b\), inconsistency means there is no \(x\) such that \(Ax = b\). 5. For example, two linear equations \(a_1x + b_1y = c_1\) and \(a_2x + b_2y = c_2\) are inconsistent if \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\). 6. This ratio condition means the lines are parallel but have different intercepts, so they never meet. 7. In summary, a system is inconsistent when it has no common solution due to contradictory equations.