SOLVE THE FOLLOWING SYSTEM OF EQUATIONS:
х-у-2z =-1
x + y + z = 6
x+ y-z= 4

We are given the system of equations:

\[
\begin{aligned}
1) \ x - y - 2z &= -1 \\
2) \ x + y + z &= 6 \\
3) \ x + y - z &= 4 \\
\end{aligned}
\]

### Step 1: Eliminate one variable (let's eliminate \(y\)).

Add equations (2) and (3):
\[
(x + y + z) + (x + y - z) = 6 + 4
\]
\[
2x + 2y = 10 \quad \Rightarrow \quad x + y = 5 \quad \Rightarrow \quad y = 5 - x
\]

### Step 2: Substitute \(y = 5 - x\) into another equation.

Substitute into equation (1):
\[
x - (5 - x) - 2z = -1
\]
Simplify:
\[
x - 5 + x - 2z = -1 \quad \Rightarrow \quad 2x - 5 - 2z = -1 \quad \Rightarrow \quad 2x - 2z = 4 \quad \Rightarrow \quad x - z = 2 \quad \Rightarrow \quad z = x - 2
\]

### Step 3: Substitute \(z = x - 2\) into \(x + y + z = 6\).

Using \(y = 5 - x\) and \(z = x - 2\), substitute into equation (2):
\[
x + (5 - x) + (x - 2) = 6
\]
Simplify:
\[
x + 5 - x + x - 2 = 6 \quad \Rightarrow \quad x + 3 = 6 \quad \Rightarrow \quad x = 3
\]

### Step 4: Solve for \(y\) and \(z\).

Now that \(x = 3\):
- \(y = 5 - x = 5 - 3 = 2\)
- \(z = x - 2 = 3 - 2 = 1\)

### Final Solution:
The solution to the system of equations is:
\[
x = 3, \quad y = 2, \quad z = 1
\]

Would you like further explanation or have any questions?

### Related questions:
1. What method can be used to solve a system of equations with more variables?
2. How does substitution differ from elimination when solving systems of equations?
3. Can matrix methods like Gaussian elimination solve this system faster?
4. How would the solution change if one equation was slightly altered?
5. What happens if there are no unique solutions (infinite solutions or none)?

### Tip:
When solving systems of equations, it's often helpful to substitute or eliminate variables systematically to simplify the system step by step.

SOLVE THE FOLLOWING SYSTEM OF EQUATIONS: х-у-2z =-1, x + y + z = 6, x+ y-z= 4

Learn how to solve a system of linear equations involving three variables with a step-by-step solution. This detailed guide uses substitution and elimination methods to solve the system: х - у - 2z = -1, x + y + z = 6, and x + y - z = 4. Ideal for students in Grades 9-12.

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