Show that the system is linear?

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SUMMARY

The system defined by the equation y(n) = x(n) + 0.8y(n-1) is not linear due to its dependence on the previous output, y(n-1). To demonstrate linearity, one must show that the system satisfies the principles of superposition and homogeneity. The proposed method of inputting a sinusoidal signal to observe a scaled and phase-shifted output is a valid approach to verify linear time-invariance (LTI). The manipulation of the equation to express y(n) in terms of x(n) and its previous values can aid in understanding the system's behavior.

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Homework Statement


Is the following system linear:
y(n) = x(n) + 0.8y(n-1)

The Attempt at a Solution


I'm really stuck on this one.

My idea was to assume that the system is casual, find the impulse response and then use the fact that: y(n)=x(n)*h(n). However, if I assume that the output is the convolution of the impulse response and the input, I have ALREADY ASSUMED (and not proven) that the system is linear.

So I am lost.

I can't wrap my brain around the fact that the output depends on the input and the previous output.

Another idea I had is that if I show that I input a sinusoid and I get a scaled and phase shifted sinusoid, then the system is LTI.

Any help?
 
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What if, assuming y(0) = x(0), you wrote:

y(n) = x(n) + 0.8y(n-1) = x(n) + 0.8x(n-1)... + (0.8)^n x(0),

would that help?
 

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