Final answer:
The system described by the equation y[n] = e^n x[n] is non-linear because it does not satisfy the superposition principle.
Step-by-step explanation:
To determine whether the system described by the equation y[n] = e^n x[n] is linear, we need to check if it satisfies two properties: superposition and homogeneity. For superposition, if we take two inputs x1[n] and x2[n] and their corresponding outputs y1[n] = e^n x1[n] and y2[n] = e^n x2[n], then the system must satisfy y[n] = y1[n] + y2[n] for the input x[n] = x1[n] + x2[n]. For homogeneity, the system must satisfy y[n] = A y1[n] for the input x[n] = A x1[n], where A is a constant. In this case, the system y[n] = e^n x[n] does not satisfy the superposition principle because if you take the sum of two inputs, x[n] = x1[n] + x2[n], the output does not equal the sum of the individual outputs for the inputs x1[n] and x2[n]. The output would be y[n] = e^n (x1[n] + x2[n]), which is not the same as e^n x1[n] + e^n x2[n]. Therefore, the system is non-linear as it violates the superposition principle.