The general case:
    The two fundamental equations:
    (1) AE = C + I + G + (X-M)
    (2) Equil.: AE = Y
    The other details:
    (3) DY = Y - T
    (4) T = To + tY, where
        To=lump-sum taxes (such as property taxes and DMV fees) and
        t = the income tax rate
    (5) C = Ca + MPC·DY
    (6) I = Ia
    (7) G = Ga
    (8) X-M = (X-M)a

Step 1 (expressing AE as a function of Y and A):
    AE = Ca + MPC·DY + Ia + Ga + (X-M)a
The trick is to replace DY with Y. Fortunately, (3) and (4) allow us to do that:
    DY = Y - T = Y - (To + tY) = Y - To - tY = -To + (1-t)Y.
So replacing DY with -To + (1-t)Y, we get:
    AE = Ca + MPC·(-To + (1-t)Y + Ia + Ga + (X-M)a
which multiplies out to:
    AE = Ca - MPC·To + MPC(1-t)Y + Ia + Ga + (X-M)a.
Now AE is a function of Y and the various components of autonomous spending.

Step 2 (setting Y = AE for the equilibrium condition):
    Y = Ca - MPC·To + MPC(1-t)Y + Ia + Ga + (X-M)a.

Step 3 (rearranging terms to solve for Y):
    Y - MPC(1-t)Y = Ca - MPC·To + Ia + Ga + (X-M)a
    (1 - MPC·(1-t))Y = Ca - MPC·To + Ia + Ga + (X-M)a

             1
    Y = -------------·[Ca - MPC·To + Ia + Ga + (X-M)a]
        1 - MPC·(1-t)

    This is the general formula for what we have been using all along:
    the respending ratio, RR = MPC·(1-t)
    the spending multiplier, m  = 1/(1-MPC·(1-t))
   and autonomous spending, A =  [Ca - MPC·To + Ia + Ga + (X-M)a]

    So as always, the general solution is: Y = m·A.
    Note1: all ordinary autonomous spending gets multiplied by the spending multiplier (m).
    Note2: lump-sum taxes (To) get multiplied by the negative of m·MPC. m·MPC is called the tax multiplier. (1) Its sign is negative because tax increases lower income and vice-versa. (2) tax changes have a smaller multiplier effect (m·MPC < m since MPC < 1) because for each $1 tax cut, you spend only MPC of it; you put the rest goes into savings.