SOLUTION: Solve the following equations by Gauss - Elimination Method : 54x+y+z=110 2x + 15y + 6z = 72 — x + 6y + 27z = 85.

Algebra ->  Equations -> SOLUTION: Solve the following equations by Gauss - Elimination Method : 54x+y+z=110 2x + 15y + 6z = 72 — x + 6y + 27z = 85.      Log On


   

Question 442505: Solve the following equations by
Gauss - Elimination Method :
54x+y+z=110
2x + 15y + 6z = 72
— x + 6y + 27z = 85.
Answer by swincher4391(1107) About Me  (Show Source):
You can put this solution on YOUR website!
Might this be linear algebra? If so, you can make an augmented matrix of coefficients.
54 1 1 | 110
2 15 6| 72
-1 6 27| 85
Take 2 *row 3 and add it to row 2.
54 1 1 | 110
0 27 60|242
-1 6 27 | 85
Switch row 1 and row 3
-1 6 27 | 85
0 27 60 | 242
54 1 1 | 110
Multiply row 1 by 54 and add it to row 3.
-1 6 27 | 85
0 27 60 | 242
0 325 1459 | 4700
Multiply row 1 by -1.
1 -6 -27 | -85
0 27 60 | 242
0 325 1459| 4700
Multiply row 2 by -12 and add it to row 3.
1 -6 -27 | -85
0 27 60 | 242
0 1 739 | 1796
Switch row 2 and 3.
1 -6 -27 | -85
0 1 739 | 1796
0 27 60 | 242
Multiply row 2 by 6 and add it to row 1.
1 0 4407 | 10691
0 1 739 | 1796
0 27 60 | 242
Multiply -27 to row 1 and add it to row 3
1 0 4407 | 10691
0 1 739 | 1796
0 0 -19893 | -42310
1 0 4407 | 10691
0 1 739 | 1796
0 0 1 | 48250/19893
Row 3 * -739 + row 2.
1 0 4407 | 10691
0 1 0 | 71708/19893
0 0 1 | 48250/19893
Row 3 * -4407 + row 1
1 0 0 | 38313 / 19893
0 1 0 | 71708 / 19893
0 0 1 | 48250 / 19893
FUN!
So our answers are x+=+38313+%2F+19893 y=71708+%2F+19893 z=48250%2F19893
Let me offer an easier way to do this. We will use this as the check.
Use Cramer's rule to find these solutions.

54 1 1
2 15 6
-1 6 27
Find the det(A) = 19893
Find the adjoint(A)
You must find each cofactor.
A11 = 15(27) - 6(6) = 405 -36 = 369 * (-1)^(1+1) = 369
A12 = 2(7) - 6(-1) = 14 +6 = 20 * (-1)^(1+2) = -20
A13 = 2(6) - 15(-1) = 12 +15 = 27 * (-1)(1+3) = 27
A21 = 27 -6 = 21 * -1 = -21
A22 = 54(27) +1 = 1459
A23 = 54(6) + 1 = 325 * -1 = -325
A31 = 6 - 15 = -9
A32 = 54(6) - 2 = 322 * -1 = -322
A33 = 54(15) -2 = 808
so our cofactor matrix is
369 -20 27
-21 1459 -325
-9 -322 808
Take its transpose.
369 -21 -9
-20 1459 -322
27 - 325 808
Divide by the determinant of A
1/19893 * adj(A)
adj(A) * solution matrix
369 -21 -9 110
-60 1459 -322 * 72
27 -325 808 85
369*110 + -21*72 + 85*-9 = 38313
-60*110 + 1459 *72 + -322*85 = 71708
27 * 110 + -325 * 72 + 808 * 85 = 48250
Divide by the determinant and we get our answer again.
And yes... I just did that by hand. :(
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I just made an edit to the work:
Now I have checked my solutions, they are now correct. I had a -20 instead of a -60 in my A21 spot.
Please make note of this.