We are asked to solve the system of equations using Gaussian elimination (or Gauss-Jordan elimination.) The system:

3x-2y+z=3

x+3y-4z=-7

2x-3y+5z=8

x-8y+9z=17

The idea is to place the coefficients into an augmented matrix. Then using basic row operations we put the matrix into reduced row echelon form (the first nonzero element in each row is a 1, and under each 1 the rest of the column is zeros.)

The row operations: we can swap rows, we can replace a row with a scalar multiple of its entries, and we can replace a row with the sum/difference of multiples of rows.

`([3,-2,1,|,3],[1,3,-4,|,-7],[2,-3,5,|,8],[1,-8,9,|,17])`

This is the augmented matrix for the system. Let the rows be designated R1 (the first row), R2, R3, and R4 (the last row.) Now swap R1 and R2, and replace R2 with -3R1 + R2, replace R3 with -2R1 + R3, and replace R4 with R1-R3 to get:

`([1,3,-4,|,-7],[0,-11,13,|,24],[0,-9,13,|,22],[0,11,-13,|,-24])`

Replace R2 with -1/11 * R2 and R4 with R2 + R4:

`([1,3,-4,|,-7],[0,1,-13/11,|,-24/11],[0,-9,13,|,22],[0,0,0,|,0])`

Replace R3 with 9R2 + R3:

`([1,3,-4,|,-7],[0,1,-13/11,|,-24/11],[0,0,26/11,|,26/11],[0,0,0,|,0])`

Replace R3 with 11/26 * R3:

`([1,3,-4,|,-7],[0,1,-13/11,|,-24/11],[0,0,1,|,1],[0,0,0,|,0])`

Replace R2 with 13/11R3 + R2:

`([1,3,-4,|,-7],[0,1,0,|,-1],[0,0,1,|,1],[0,0,0,|,0])`

Replace R1 with -3R2 + R1:

`([1,0,-4,|,-4],[0,1,0,|,-1],[0,0,1,|,1],[0,0,0,|,0])`

Replace R1 with 4R3 + R1:

`([1,0,0,|,0],[0,1,0,|,-1],[0,0,1,|,1],[0,0,0,|,0])`

Thus x=0, y=-1, z=1.

Note that the order of row operations is not important. You can do any permissible row operations at any time, though there may be more efficient procedures.

**Further Reading**

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