Solve for the variables in these simultaneous equations: 3x- 2y - z = -8, 5x + y + 3z = 23 and 4x + y - 5z = -18 ?

To solve these simultaneous equations, we can use the method of substitution or elimination. Let's use the elimination method:

Given equations:
1) 3x - 2y - z = -8
2) 5x + y + 3z = 23
3) 4x + y - 5z = -18

First, let's add equations 1 and 3 to eliminate y:
(3x - 2y - z) + (4x + y - 5z) = -8 + (-18)
7x - 6z = -26
7x = 6z - 26
x = (6z - 26) / 7

Now, substitute x into equation 1:
3((6z - 26) / 7) - 2y - z = -8
18z/7 - 78/7 - 2y - z = -8
18z - 78 - 14y - 7z = -56
11z - 14y = 22
11z = 14y + 22
z = (14y + 22) / 11

Now, substitute x and z into equation 2:
5((6z - 26) / 7) + y + 3z = 23
30z/7 - 130/7 + y + 3z = 23
30z - 130 + 7y + 21z = 161
51z + 7y = 291
51(14y + 22) / 11 + 7y = 291
714y + 1122 + 77y = 3201
791y = 2079
y = 2079 / 791
y = 3

Now, substitute y back into the equations to find x and z:
x = (6z - 26) / 7
z = (14(3) + 22) / 11

x = (6z - 26) / 7
x = (6(8) - 26) / 7
x = (48 - 26) / 7
x = 22 / 7
x = 3.14

z = (14(3) + 22) / 11
z = (42 + 22) / 11
z = 64 / 11
z = 5.82

Therefore, the solution to the simultaneous equations is x = 3.14, y = 3, and z = 5.82.