y[n]=0.25x[n]+0.25x[n+1]+0.25x[n-1]+0.25x[n+2]

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Solution for y[n]=0.25x[n]+0.25x[n+1]+0.25x[n-1]+0.25x[n+2] equation: 


Simplifying
y[n] = 0.25x[n] + 0.25x[n + 1] + 0.25x[n + -1] + 0.25x[n + 2]

Multiply y * n
ny = 0.25x[n] + 0.25x[n + 1] + 0.25x[n + -1] + 0.25x[n + 2]

Multiply x * n
ny = 0.25nx + 0.25x[n + 1] + 0.25x[n + -1] + 0.25x[n + 2]

Reorder the terms:
ny = 0.25nx + 0.25x[1 + n] + 0.25x[n + -1] + 0.25x[n + 2]
ny = 0.25nx + [1 * 0.25x + n * 0.25x] + 0.25x[n + -1] + 0.25x[n + 2]

Reorder the terms:
ny = 0.25nx + [0.25nx + 0.25x] + 0.25x[n + -1] + 0.25x[n + 2]
ny = 0.25nx + [0.25nx + 0.25x] + 0.25x[n + -1] + 0.25x[n + 2]

Reorder the terms:
ny = 0.25nx + 0.25nx + 0.25x + 0.25x[-1 + n] + 0.25x[n + 2]
ny = 0.25nx + 0.25nx + 0.25x + [-1 * 0.25x + n * 0.25x] + 0.25x[n + 2]

Reorder the terms:
ny = 0.25nx + 0.25nx + 0.25x + [0.25nx + -0.25x] + 0.25x[n + 2]
ny = 0.25nx + 0.25nx + 0.25x + [0.25nx + -0.25x] + 0.25x[n + 2]

Reorder the terms:
ny = 0.25nx + 0.25nx + 0.25x + 0.25nx + -0.25x + 0.25x[2 + n]
ny = 0.25nx + 0.25nx + 0.25x + 0.25nx + -0.25x + [2 * 0.25x + n * 0.25x]

Reorder the terms:
ny = 0.25nx + 0.25nx + 0.25x + 0.25nx + -0.25x + [0.25nx + 0.5x]
ny = 0.25nx + 0.25nx + 0.25x + 0.25nx + -0.25x + [0.25nx + 0.5x]

Reorder the terms:
ny = 0.25nx + 0.25nx + 0.25nx + 0.25nx + 0.25x + -0.25x + 0.5x

Combine like terms: 0.25nx + 0.25nx = 0.5nx
ny = 0.5nx + 0.25nx + 0.25nx + 0.25x + -0.25x + 0.5x

Combine like terms: 0.5nx + 0.25nx = 0.75nx
ny = 0.75nx + 0.25nx + 0.25x + -0.25x + 0.5x

Combine like terms: 0.75nx + 0.25nx = 1nx
ny = 1nx + 0.25x + -0.25x + 0.5x

Combine like terms: 0.25x + -0.25x = 0.00
ny = 1nx + 0.00 + 0.5x
ny = 1nx + 0.5x

Solving
ny = 1nx + 0.5x

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '-1nx' to each side of the equation.
-1nx + ny = 1nx + -1nx + 0.5x

Combine like terms: 1nx + -1nx = 0
-1nx + ny = 0 + 0.5x
-1nx + ny = 0.5x

Combine like terms: 0.5x + -0.5x = 0.0
-1nx + ny + -0.5x = 0.0

The solution to this equation could not be determined.

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