To solve the system of equations given:
1. **Write the system of equations:**
- x + 2y + z = 4
- 2x - y + 4z = -8
- -3x + y - 2z = -1
2. **Choose a method to solve the system:**
- One common method is to use the elimination method. We'll eliminate one variable at a time to solve for the others.
3. **Eliminate a variable:**
- Start by eliminating a variable between two pairs of equations. Let's eliminate y from the first and third equations:
- From the first equation: y = 4 - x - z
- Substitute y in the third equation: -3x + (4 - x - z) - 2z = -1
- Simplify: -4x - 3z = -1
4. **Eliminate another variable:**
- Now, eliminate y from the second and third equations:
- From the second equation: y = 2x + 4z + 8
- Substitute y in the third equation: -3x + (2x + 4z + 8) - 2z = -1
- Simplify: -x + 2z = -9
5. **Solve the system of equations:**
- Now, you have two equations:
- -4x - 3z = -1
- -x + 2z = -9
- Solve these two equations simultaneously to find the values of x and z.
6. **Find the values of x, y, and z:**
- Once you have x and z, substitute them back into one of the original equations to solve for y.
7. **Check your solution:**
- Finally, check your solution by substituting x, y, and z into all three original equations to ensure they satisfy all of them.
By following these steps, you should be able to solve the system of equations and find the values of x, y, and z that satisfy all three equations simultaneously.
