Find the solution (x, y, z) for the system of equations:
5x + 4y - 8z = 1
7x - 9y + z = -1
2x + 3y - 4z = 1

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We have the system of equations:
1) 5x + 4y - 8z = 1
2) 7x - 9y + z = -1
3) 2x + 3y - 4z = 1

Step 1: From equation (2), express z in terms of x and y:
7x - 9y + z = -1 => z = -1 - 7x + 9y

Step 2: Substitute z into equations (1) and (3):

Equation (1):
5x + 4y - 8(-1 - 7x + 9y) = 1
=> 5x + 4y + 8 + 56x - 72y = 1
=> (5x + 56x) + (4y - 72y) + 8 = 1
=> 61x - 68y + 8 = 1
=> 61x - 68y = 1 - 8 = -7  ...(4)

Equation (3):
2x + 3y - 4(-1 - 7x + 9y) = 1
=> 2x + 3y + 4 + 28x - 36y = 1
=> (2x + 28x) + (3y - 36y) + 4 = 1
=> 30x - 33y + 4 = 1
=> 30x - 33y = 1 - 4 = -3  ...(5)

Step 3: Solve the two equations (4) and (5):
61x - 68y = -7
30x - 33y = -3

Multiply (5) by 2:
60x - 66y = -6

Subtract from (4):
(61x - 68y) - (60x - 66y) = -7 - (-6)
=> (61x - 60x) + (-68y + 66y) = -7 + 6
=> x - 2y = -1
=> x = -1 + 2y  ...(6)

Substitute x into (5):
30(-1 + 2y) - 33y = -3
=> -30 + 60y - 33y = -3
=> 27y - 30 = -3
=> 27y = 27
=> y = 1

Step 4: Substitute y = 1 into (6):
x = -1 + 2(1) = -1 + 2 = 1

Step 5: Substitute x = 1 and y = 1 into expression for z:
z = -1 - 7(1) + 9(1) = -1 - 7 + 9 = 1

Therefore, the solution is x = 1, y = 1, z = 1.

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