Linear and time-varying system

Introduction:
In electrical engineering, systems can be classified based on their linearity and time-invariance properties. Linearity refers to the property of a system where the output is directly proportional to the input, and time-invariance refers to the property where the system's response does not change with time.

Explanation:
The given system is described by the relation: y(t) = tx(t). Let's analyze this system based on the properties of linearity and time-invariance.

Linearity:
A system is linear if it satisfies two properties: superposition and scaling.

1. Superposition: The system must satisfy the principle of superposition, which means that if two inputs x1(t) and x2(t) produce outputs y1(t) and y2(t) respectively, then the input x(t) = x1(t) + x2(t) should produce the output y(t) = y1(t) + y2(t).

In the given system, let's consider two inputs x1(t) and x2(t) that produce outputs y1(t) and y2(t) respectively.
For x1(t): y1(t) = tx1(t)
For x2(t): y2(t) = tx2(t)

Now, let's consider the input x(t) = x1(t) + x2(t):
y(t) = tx(t)
= t(x1(t) + x2(t))
= tx1(t) + tx2(t)

Comparing this with the output y(t) = y1(t) + y2(t), we can see that the system satisfies the principle of superposition. Therefore, the system is linear with respect to superposition.

2. Scaling: The system must also satisfy the property of scaling, which means that if an input x(t) produces an output y(t), then scaling the input by a constant a should produce a scaled output ay(t).

In the given system, let's consider an input x(t) that produces an output y(t):
y(t) = tx(t)

Now, let's scale the input by a constant a:
y1(t) = a(tx(t))

Comparing this with the output ay(t), we can see that the system satisfies the property of scaling. Therefore, the system is linear with respect to scaling.

Since the system satisfies both the properties of linearity (superposition and scaling), it is classified as a linear system.

Time-invariance:
A system is time-invariant if its response does not change with time. In other words, if a time-shifted input x(t - τ) produces a time-shifted output y(t - τ), then the system is time-invariant.

In the given system, let's consider a time-shifted input x(t - τ) that produces a time-shifted output y(t - τ):
y(t - τ) = (t - τ)x(t - τ)

Comparing this with the output y(t) = tx(t), we can see that the system's response changes with time due to the presence of the time-shifted variable τ. Therefore, the system is not time-invariant.

Conclusion:
Based on the analysis, the given system is linear (satisfies the properties of superposition and scaling) and time-v