Final answer:
An equation has no solutions when it is contradictory or inconsistent. This occurs when the variables cancel out and the equation simplifies to a false statement, or when there are conflicting equations that cannot be simultaneously true.
Step-by-step explanation:
An equation has no solutions when the equation is contradictory or inconsistent. This means that there are no values of the unknown variable(s) that satisfy the equation. One condition for an equation to have no solutions is when the variables cancel out and the equation simplifies to a false statement, such as 2 = 5. Another condition is when there are conflicting equations that cannot be simultaneously true. For example, if one equation states that x = 3 and another equation states that x = 2, there is no value of x that can satisfy both equations simultaneously. In both cases, the system of equations is considered inconsistent and has no solution.