Alright, let's start by writing out the equations again:
1. x - y + z = -3,
2. -3x + 4y - z = 2,
3. x - 3y - 2z = 7.
We will use the method of substitution to solve this system. This involves solving one of the equations for one variable in terms of the other variables and substituting this into the other equations.
Since the coefficient of x in the first equation and the third equation is 1, these equations are good starting points.
For the sake of presentation, let's rewrite these equations:
Let's wave our magic wand and solve the equations:
From the first equation: x = y - z - 3 ... (a)
From the third equation: x = 3y + 2z - 7 ... (b)
Since x is the same in both equations a and b, we can equate them and solve for z:
y - z - 3 = 3y + 2z - 7
=> 2y - 3z = 4
=> z = 2 - 2/3*y ... (c)
Now substitute z from equation (c) into equation (a):
x = y - (2 - 2/3*y) - 3 = 2/3*y - 5 ... (d)
Now substitute y and z from equations (c) and (d) into the second equation:
-12/3 + 16/3 + 2 = 2, which checks out true.
Therefore, the solutions are:
x = 2 (from equation (d))
y = 1 (from equation (c))
z = -4 (from equation (c))
This is your solution set to the system of equations: {2, 1, -4}.