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

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

From the first equation, we can express y in terms of x and z:
y = 23 - 5x - 3z

Substitute this expression for y in the second equation:
4x + (23 - 5x - 3z) - 5z = -18
4x + 23 - 5x - 3z - 5z = -18
-x - 8z = -41
x = 41 - 8z

Now substitute the expression for x in terms of z into the third equation:
3(41 - 8z) - 2y - z = -8
123 - 24z - 2y - z = -8
123 - 25z - 2y = -8
-2y = 25z + 115
y = -12.5z - 57.5

Now we have expressions for x, y, and z in terms of z. Let's substitute these back into the original equations to find the values of x, y, and z.

From the first equation:
5(41 - 8z) + (-12.5z - 57.5) + 3z = 23
205 - 40z - 12.5z - 57.5 + 3z = 23
205 - 49.5z - 57.5 + 3z = 23
147.5 - 46.5z = 23
-46.5z = -124.5
z = 2.68

Now substitute z = 2.68 into the expressions for x and y:
x = 41 - 8(2.68) = 20.64
y = -12.5(2.68) - 57.5 = -90.5

Therefore, the solution to the simultaneous equations is:
x = 20.64, y = -90.5, z = 2.68