We are given the system of equations:

$\text{Equation 1: } x - y + z = 1$ $\text{Equation 2: } 2x - y - 2z = 3$ $\text{Equation 3: } x + 3y - 3z = -1$

Step 1: Eliminate one variable (let's eliminate $y$)

First, subtract Equation 1 from Equation 2 to eliminate $y$:

$(2x - y - 2z) - (x - y + z) = 3 - 1$ Simplifying: $2x - y - 2z - x + y - z = 2$ $x - 3z = 2 \quad \text{(Equation 4)}$

Next, subtract Equation 1 from Equation 3 to eliminate $y$:

$(x + 3y - 3z) - (x - y + z) = -1 - 1$ Simplifying: $x + 3y - 3z - x + y - z = -2$ $4y - 4z = -2$ Dividing by 2: $2y - 2z = -1 \quad \text{(Equation 5)}$

Step 2: Solve the system of two equations

Now we have the reduced system:

1. $x - 3z = 2$ (Equation 4)
2. $2y - 2z = -1$ (Equation 5)

Solve for $x$ in terms of $z$ from Equation 4:

$x = 3z + 2$

Solve for $y$ in terms of $z$ from Equation 5:

$2y = 2z - 1$ $y = z - \frac{1}{2}$

Step 3: Substitute into the original equation

Now that we have $x = 3z + 2$ and $y = z - \frac{1}{2}$, substitute these into Equation 1 to find $z$:

$x - y + z = 1$ $(3z + 2) - \left(z - \frac{1}{2}\right) + z = 1$ Simplifying: $3z + 2 - z + \frac{1}{2} + z = 1$ $3z + \frac{5}{2} = 1$ $3z = 1 - \frac{5}{2}$ $3z = -\frac{3}{2}$ $z = -\frac{1}{2}$

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

Now that we know $z = -\frac{1}{2}$, we can substitute this value into the expressions for $x$ and $y$:

$x = 3z + 2 = 3\left(-\frac{1}{2}\right) + 2 = -\frac{3}{2} + 2 = \frac{1}{2}$ $y = z - \frac{1}{2} = -\frac{1}{2} - \frac{1}{2} = -1$

Final Solution:

$x = \frac{1}{2}, \quad y = -1, \quad z = -\frac{1}{2}$

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How can substitution be used to solve this system?
2. What happens if you choose to eliminate $z$ instead of $y$?
3. How does elimination compare to solving systems graphically?
4. Can this method be applied to larger systems of equations?
5. What real-world problems can be modeled by systems of equations?

Tip: Always double-check each step of elimination to avoid sign errors!