# Linear and time-invariant system

## Homework Statement

consider these systems is linear or time invariant?
a) Y[n]=aX[n]+b
b) Y[n]=X[n] X[n-5]
c) Y[n]=5X[n]+9X[n-5]

## Homework Equations

linearity of the system is defined by the principle of proportionality and superposition
and time-invariant system means if a time shift of the input sequence causes a coresponding shift in the output sequence.

## The Attempt at a Solution

a) Y[n]=aX[n]+b
input X[n]
X1[n]= X[n-no]
Y1[n]= aX[n-no]+b
Y[n-no]= aX[n-no]+b
so,this system is time-invariant

b)Y[n]=X[n] X[n-5]
input X[n]
Y1[n]=X[n-no] X[n-no-5]
Y[n-no]=X[n-no] X[n-no-5] <- here I'm not sure..
but, I think this system is time-invariant

c)Y[n]=5X[n]+9X[n-5]
input X[n]=X[n-no]
Y1[n]=5X[n-no]+9X[n-no-5]
Y[n-no]=5X[n]-5X[no]+9X[n]-9X[no]+9X[-5]
since the output wasn't as we expect, so the system is not time-invariant

can, someone check my works, am I correct or wrong? and also, how to check whether the system is linear, I know that theoretically the linear system should satisfy the superposition and proportionality, but I'm not sure how to work out with it.. thanks