To solve this system of three equations with three variables ($x$, $y$, and $z$), we can use substitution or elimination. The equations are:

1. $2x + y - 2z = 4$
2. $x + y = 1$
3. $3x + 2y - z = 2$

Step 1: Simplify and Substitute from $x + y = 1$

From Equation (2): $y = 1 - x$

Step 2: Substitute $y = 1 - x$ into the other two equations.

Substituting into Equation (1):

$2x + (1 - x) - 2z = 4$ Simplify: $x + 1 - 2z = 4$ x - 2z = 3 \tag{4}

Substituting into Equation (3):

$3x + 2(1 - x) - z = 2$ Simplify: $3x + 2 - 2x - z = 2$ x - z = 0 \tag{5}

Step 3: Solve the two new equations ($x - 2z = 3$ and $x - z = 0$).

From Equation (5): x = z \tag{6}

Substitute $x = z$ into Equation (4): $z - 2z = 3$ $-z = 3$ $z = -3$

Step 4: Solve for $x$ and $y$.

Using $z = -3$ in Equation (6): $x = -3$

Using $x = -3$ in $y = 1 - x$: $y = 1 - (-3)$ $y = 4$

Final Solution:

$x = -3, \, y = 4, \, z = -3$

Verification:

Substitute $x = -3$, $y = 4$, and $z = -3$ back into the original equations to confirm:

1. $2(-3) + 4 - 2(-3) = -6 + 4 + 6 = 4$ ✅
2. $(-3) + 4 = 1$ ✅
3. $3(-3) + 2(4) - (-3) = -9 + 8 + 3 = 2$ ✅

The solution satisfies all equations.

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Let me know if you'd like further explanation! Here are 5 related questions you might find useful:

1. How do elimination and substitution methods compare in solving such systems?
2. Can systems of equations always be solved? What conditions make them unsolvable or have infinitely many solutions?
3. What are matrix methods (e.g., Gaussian elimination) to solve systems of equations?
4. How does graphing help visualize the solution of such systems?
5. Can this problem be extended to four or more variables? If so, how?

Tip: When solving systems of equations, substituting variables early can simplify the system quickly.