We are tasked with solving the following system of equations:

1. $x + y + z = 2$
2. $x + 2y - z = 6$
3. $2x + y - z = 5$

We can solve this system using substitution or elimination. Let's go step by step.

Step 1: Eliminate one variable

Let's eliminate $z$ from two of the equations. We can do this by adding equations (1) and (2).

Add Equation (1) and Equation (2):

$(x + y + z) + (x + 2y - z) = 2 + 6$ $2x + 3y = 8 \quad \text{(Equation 4)}$

Now, add Equation (1) and Equation (3):

$(x + y + z) + (2x + y - z) = 2 + 5$ $3x + 2y = 7 \quad \text{(Equation 5)}$

Step 2: Solve the new system of two equations

We now have the system of two equations:

$2x + 3y = 8 \quad \text{(Equation 4)}$ $3x + 2y = 7 \quad \text{(Equation 5)}$

We can solve this system by elimination or substitution. Let's multiply Equation (4) by 3 and Equation (5) by 2 to align the coefficients of $x$ for elimination.

Multiply Equation (4) by 3:

$6x + 9y = 24 \quad \text{(Equation 6)}$

Multiply Equation (5) by 2:

$6x + 4y = 14 \quad \text{(Equation 7)}$

Step 3: Eliminate $x$

Now subtract Equation (7) from Equation (6):

$(6x + 9y) - (6x + 4y) = 24 - 14$ $5y = 10$ $y = 2$

Step 4: Substitute $y = 2$ into one of the original equations

Now substitute $y = 2$ into Equation (4) or (5). Let's use Equation (5):

$3x + 2(2) = 7$ $3x + 4 = 7$ $3x = 3$ $x = 1$

Step 5: Solve for $z$

Substitute $x = 1$ and $y = 2$ into Equation (1):

$1 + 2 + z = 2$ $3 + z = 2$ $z = -1$

Final Solution:

The solution to the system of equations is:

$x = 1, \quad y = 2, \quad z = -1$

Let me know if you'd like further details or have any questions.

Follow-up questions:

1. How would you solve a system of nonlinear equations?
2. What is the graphical method of solving systems of equations?
3. Can this system have infinitely many solutions?
4. How do you check the solution of a system of equations?
5. What other methods are available to solve systems of equations apart from substitution and elimination?

Tip: When solving systems of equations, always look for opportunities to eliminate variables early to simplify the process.