# Solve the equations x - y + z = 6, 3x - 3y + z = 9 and x + 2y - z = 4

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You may use determinants as alternative method such that:

`Delta = [[1,-1,1],[3,-3,1],[1,2,-1]]`

`Delta = 3 + 6 - 1 + 3 - 2 - 3 =gt Delta = 6`

`x = (Delta_x)/Delta`

`Delta_x = [[6,-1,1],[9,-3,1],[4,2,-1]]`

`Delta_x = 18 + 18 - 4 + 12 - 12 - 9 =gt Delta_x = 23`

`x= 23/6 `

`Delta_y = [[1,6,1],[3,9,1],[1,4,-1]]`

`Delta_y = -9 + 12 + 6 - 9 + 18 - 4 =gt Delta_y = 14`

`y = (Delta_y)/Delta`

`y = 14/6 =gt y = 7/3`

`Delta_z = [[1,-1,6],[3,-3,9],[1,2,4]]`

`Delta_z = -12 + 36 - 9 + + 18 - 18 + 12 =gt Delta_z = 27`

`z= (Delta_z)/Delta`

z`= 27/6 =gt z = 9/2`

**Hence, evaluating the solutions to the system of three linear equations yields `x= 23/6 ; y = 7/3; z = 9/2` .**

**Sources:**

The system of equations

x - y + z = 6 ...(1)

3x - 3y + z = 9 ...(2)

x + 2y - z = 4 ...(3)

has to be solved.

3*(1) - (2)

=> 3z - z = 18 - 9 = 9

=> z = 9/2

(3) - (1)

=> 3y - 2z = -2

substitute z = 9/2

=> 3y - 9 = -2

=> 3y = 7

=> y = 7/3

substitute z = 9/2 and y = 7/3 in (1)

=> x = 6 - 9/2 + 7/3

=> x = 23/6

**The solution of the set of equations is x = 23/6, y = 7/3 and z = 9/2**