y[n]=0.5y[n-1]+x[n]

Solution for y[n]=0.5y[n-1]+x[n] equation:

Simplifying
y[n] = 0.5y[n + -1] + x[n]

Multiply y * n
ny = 0.5y[n + -1] + x[n]

Reorder the terms:
ny = 0.5y[-1 + n] + x[n]
ny = [-1 * 0.5y + n * 0.5y] + x[n]

Reorder the terms:
ny = [0.5ny + -0.5y] + x[n]
ny = [0.5ny + -0.5y] + x[n]

Multiply x * n
ny = 0.5ny + -0.5y + nx

Reorder the terms:
ny = nx + 0.5ny + -0.5y

Solving
ny = nx + 0.5ny + -0.5y

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 = nx + 0.5ny + -1nx + -0.5y

Reorder the terms:
-1nx + ny = nx + -1nx + 0.5ny + -0.5y

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

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

Combine like terms: ny + -0.5ny = 0.5ny
-1nx + 0.5ny = 0.5ny + -0.5ny + -0.5y

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

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

The solution to this equation could not be determined.