求聯立方程公式

ax + by + cz = d ...(1)
ex + fy + gz = h ... (2)
tx + uy + vz = w ... (3)
(1) => z = (d - ax - by)/c ... (4)
Sub (4) into (2), ex + fy + g(d - ax - by)/c = h
cex + cfy + dg - agx - bgy = ch
(ce - ag)x + (cf - bg)y = (ch - dg) ...(5)
Sub (4) into (3), tx + uy + v(d - ax - by)/c = w
ctx + cuy + dv - avx - bvy = cw
(ct - av)x + (cu - bv)y = (cw - dv) ... (6)
(cu - bv) * (5) - (cf - bg) * (6) =>
(cu - bv)(ce - ag)x - (cf - bg)(ct - av)x = (cu - bv)(ch - dg) - (cf - bg)(cw - dv)
(c^2eu - acgu - bcev + abgv - c^2ft + acfv + bcgt - abgv)x =
c^2hu - cdgu - bchv + bdgv - c^2fw + cdfv + bcgw - bdgv
c(ceu - agu - bev - cft + afv + bgt)x = c(chu - dgu - bhv - cfw + dfv + bgw)
x = (dfv - dgu + bgw - bhv + chu - cfw)/(afv - agu + bgt - bev + ceu - cft)
(ct - av) * (5) - (ce - ag) * (6) =>
(ct - av)(cf - bg)y - (ce - ag)(cu - bv)y = (ct - av)(ch - dg) - (ce - ag)(cw - dv)
(c^2ft - bcgt - acfv + abgv - c^2eu + bcev + acgu - abgv)y =
c^2ht - cdgt - achv + adgv - c^2ew + cdev + acgw - adgv
c(cft - bgt - afv - ceu + bev + agu)y = c(cht - dgt - ahv - cew + dev + agw)
y = (agw - ahv + dev - dgt + cht - cew)/(agu - afv + bev - bgt + cft - ceu)
y = (ahv - agw + dgt - dev + cew - cht)/(afv - agu + bgt - bev + ceu - cft)
Sub x and y into (4),
cz = d - a(dfv - dgu + bgw - bhv + chu - cfw)/(afv - agu + bgt - bev + ceu - cft) - b(ahv - agw + dgt - dev + cew - cht)/(afv - agu + bgt - bev + ceu - cft)
cz = [d(afv - agu + bgt - bev + ceu - cft) - a(dfv - dgu + bgw - bhv + chu - cfw) - b(ahv - agw + dgt - dev + cew - cht)] / (afv - agu + bgt - bev + ceu - cft)
cz = (adfv - adgu + bdgt - bdev + cdeu - cdft - adfv + adgu - abgw + abhv - achu + acfw - abhv + abgw - bdgt + bdev - bcew + bcht) / (afv - agu + bgt - bev + ceu - cft)
cz = (cdeu - cdft - achu + acfw - bcew + bcht) / (afv - agu + bgt - bev + ceu - cft)
z = (deu - dft - ahu + afw - bew + bht) / (afv - agu + bgt - bev + ceu - cft)
z = (afw - ahu + bht - bew + deu - dft) / (afv - agu + bgt - bev + ceu - cft)