A linear system is one where, if we have two inputs $x_1$ and $x_2$, producing outputs $y_1$ and $y_2$, then the output for an input of $\alpha{}x_1 + \beta{}x_2$ is $\alpha{}y_1 + \beta{}y_2$. This is precisely the property we rely on when we apply superposition to solve a problem with inputs composed of sums of easier-to-analyze inputs.

This means, essentially, that a linear system is defined to be a system in which superposition can be applied.

A linear system is one where, if we have two inputs $x_1$ and $x_2$, producing outputs $y_1$ and $y_2$, then the output for an input of $\alpha{}x_1 + \beta{}x_2$ is $\alpha{}y_1 + \beta{}y_2$. This is precisely the property we rely on when we apply superposition to solve a problem with inputs composed of sums of easier-to-analyze inputs.

This means, essentially, that if we have a system where superposition can be applied, then by definition we call that a linear system.