# Use the Gauss-Jordan method to solve the following system: -25x - 9y - z = -66 -15x - 15y - 2z = -4 20x + 9y + z = 46

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The solution of the following set of equations has to be determined using Gaussâ€“Jordan elimination:

-25x - 9y - z = -66 ...(1)

-15x - 15y - 2z = -4 ...(2)

20x + 9y + z = 46 ...(3)

This method involves the creation of a matrix with the coefficients of each of the variables, followed by conversion of the matrix to reduced row echelon form.

The matrix that created using the given equations is:

`[[-25, -9, -1,|,-66],[-15, -15,-2,|,-4],[20,9,1,|,46]]`

The reduced echelon form is arrived at as follows:

Add row 3 to row 1

=> `[[-5, 0, 0,|,-20],[-15, -15,-2,|,-4],[20,9,1,|,46]]`

Divide row 1 by -5

=> `[[1, 0, 0,|,4],[-15, -15,-2,|,-4],[20,9,1,|,46]]`

Subtract 20 times row 1 from row 3

=> `[[1, 0, 0,|,4],[-15, -15,-2,|,-4],[0,9,1,|,-34]]`

Add 15 times row 1 and twice of row 3 to row 2

=> `[[1, 0, 0,|,4],[0, 3,0,|,-12],[0,9,1,|,-34]]`

Divide row 2 by 3

=> `[[1, 0, 0,|,4],[0, 1,0,|,-4],[0,9,1,|,-34]]`

Subtract 9 times row 2 from row 3

=> `[[1, 0, 0,|,4],[0, 1,0,|,-4],[0,0,1,|,2]]`

**The matrix has been converted to the required form and the solution of the system of equations is x = 4, y = -4 and z = 2**