Solving recurrence systems

Solve the recurrence system below and dtermine a formula for the time complexity function T(n).

T(0) = 4.

T(n) = 5 + T(n - 1) (n > 0)

My effort at solving is below...

T(3) = 5 + T(2).

T(2) = 5 + T(1).

T(1) = 5 + T(0).

T(3) + T(2) + T(1) = 3 * 5 + T(2) + T(1) + T(0)

T(3) = 3 * 5 + T(0)

T(3) = 15 + 4.

Now try to obtain a formula for T(n) in terms of n.

T(n) = 5 + T(n - 1).

T(n - 1) = 5 + T(n - 2).

T(1) = 5 + T(0).

T(0) = 4.

T(n) + T(n - 1) + T(1) + T(0) = 3 * 5 + T(n - 1) + T(n - 2) + T(0) + 4.

**THIS IS WHERE MY BRAIN HITS MELTING POINT AND THE COOLANT IS RELEASED.

I'm pretty sure the formula should be T(n) = 5 * n + 4. So,

T(0) = 5 * 0 + 4 = 4

T(3) = 5 * 3 + 4 = 19.

I need to prove it though :(