x $$-$$ $$4y + 7z = g$$
      $$3y$$ $$-$$ $$5z = h$$
$$-$$$$2x + 5y$$ $$-$$ $$9z = k$$

$$D = \left| {\matrix{ 1 & { - 4} & 7 \cr 0 & 3 & { - 5} \cr { - 2} & 5 & { - 9} \cr } } \right|$$

$$D = 1\left( { - 27 + 25} \right) - 2\left( {20 - 21} \right)$$

$$D = - 2 + 2 = 0$$

If system is consistent then $${D_1} = {D_2} = {D_3} = 0$$

$$\left| {\matrix{ 1 & { - 4} & g \cr 0 & 3 & h \cr { - 2} & 5 & k \cr } } \right| = 0$$

$$1\left( {3k - 5h} \right) - 2\left( { - 4h - 3g} \right) = 0$$

$$3k - 5h + 8h + 6g = 0$$

$$6g + 3h + 3k = 0$$

$$2g + h + k = 0$$