When solving a linear system algebraically how do you know when there is no solution?

A linear system has no solution when the equations represent parallel lines that never intersect. This occurs when the ratiOS of the coefficients of the variables are equal, but the ratio of the constant terms is different. In other Words, if you can manipulate the equations to reach a contradiction (such as an equation like (0 = 1)), it indicates that the system is inconsistent and has no solution.