Mathematically, for a continuous-time system, given two arbitrary inputs x 1 ( t ) x 2 ( t ) as well as their respective zero-state outputs y 1 ( t ) = H { x 1 ( t ) } y 2 ( t ) = H { x 2 ( t ) } then a linear system must satisfy α α y 1 ( t ) + β β y 2 ( t ) = H { α α x 1 ( t ) + β β x 2 ( t ) } for any scalar values α and β, for any input signals x 1 (t) and x 2 (t), and for all time t.