<PRE><B>Question 442505</B>
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 / 19893}}} {{{y=71708 / 19893}}} {{{z=48250/19893}}}

Let me offer an easier way to do this. We will use this as the check.

Use Cramer&#39;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.
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