a) Y[n]=aX[n]+b

b) Y[n]=X[n] X[n-5]

c) Y[n]=5X[n]+9X[n-5]

I know that time-invariant system means if a time shift of the input sequence causes a coresponding shift in the output sequence.

here are 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