Question

question 5
2 pts
explain what it means if a system of equations has no solution

Explanation:

A system of equations (e.g., linear equations) has no solution when the equations represent parallel lines (in 2D) or parallel planes/hyperplanes (in higher dimensions) that never intersect. Algebraically, for a linear system \(Ax = b\), if the augmented matrix \([A|b]\) has a different rank than the coefficient matrix \(A\), or if solving leads to a contradiction (like \(0 = 5\)), there’s no solution. Geometrically, no common point satisfies all equations simultaneously.

Answer:

A system of equations has no solution when there is no set of values for the variables that satisfies all equations in the system simultaneously. For linear systems, this occurs when the equations represent parallel (non - intersecting) lines (in 2 - variable cases) or parallel planes/hyperplanes (in more variables), or algebraically when solving leads to a contradiction (e.g., \(0 = 5\)) or the rank of the coefficient matrix differs from the rank of the augmented matrix.