Write from the given conditions:
1x + 1y + 1z = 153
4x + 0y − 1z = 0
1x − 1y + 0z = -9
Construct a numerical array or "matrix" from the coefficients of x, y, & z above:
1......1.......1
4......0......-1
1.....-1.......0
Tie the numbers in this matrix to the letter positions shown below:
A......B......C
G.....H.......I
M.....N......O
Cramer's Rule from Linear Algebra will show that the
Principal Determinant of this matrix is equal to
AHO − ANI − GBO + GNC + MBI − MHC.
Place corresponding numbers into the letter positions:
(1×0×0) − (1×-1×-1) − (4×1×0) + (4×-1×1) + (1×1×-1) − (1×0×1)
goes to -6 as the Principal Determinant.
Now, write the first numerical matrix again but put the column or "stack" (153....0....-9)
in place of the first column (1....4....1):
153.........1...........1
0.............0..........-1
-9...........-1............0
Place values in this new matrix according to
AHO − ANI − GBO + GNC + MBI − MHC.
Obtain (153×0×0) − (153×-1×-1) − (0×1×0) + (0×-1×1) + (-9×1×-1) − (-9×0×1)
which gives -144.
The column (1....4....1) just replaced in the first numerical matrix consists
of all the coefficients of x in the equations at the top of this solution. The
value of the first number x is then given by the x-determinant divided by
the principal determinant or -144/-6 or 24.
From x = y − 9, take y = 24 + 9 or 33.
From z = 4x, take z = 4(24) or 96.
