To solve the system of equations:

1. \( 2x + y - 3z = 3 \) \quad (Equation 1)
2. \( x + y + 3z = 2 \) \quad (Equation 2)
3. \( 3x - 2y + z = -1 \) \quad (Equation 3)

Let's use the method of elimination or substitution. We'll start by manipulating the equations to eliminate variables.

### Step 1: Solve for one variable

Let’s solve Equation 2 for \( y \):

\[
y = 2 - x - 3z \quad \text{(Equation 4)}
\]

### Step 2: Substitute into other equations

Now, substitute Equation 4 into Equations 1 and 3:

**Substituting into Equation 1:**

\[
2x + (2 - x - 3z) - 3z = 3
\]
Simplifying:

\[
2x + 2 - x - 3z - 3z = 3
\]
\[
x - 6z + 2 = 3
\]
\[
x - 6z = 1 \quad \text{(Equation 5)}
\]

**Substituting into Equation 3:**

\[
3x - 2(2 - x - 3z) + z = -1
\]
Simplifying:

\[
3x - 4 + 2x + 6z + z = -1
\]
\[
5x + 7z - 4 = -1
\]
\[
5x + 7z = 3 \quad \text{(Equation 6)}
\]

### Step 3: Solve the new system of equations

Now we have a new system of equations to solve:

1. \( x - 6z = 1 \) \quad (Equation 5)
2. \( 5x + 7z = 3 \) \quad (Equation 6)

From Equation 5, we can express \( x \) in terms of \( z \):

\[
x = 1 + 6z \quad \text{(Equation 7)}
\]

### Step 4: Substitute into Equation 6

Substitute Equation 7 into Equation 6:

\[
5(1 + 6z) + 7z = 3
\]

Expanding:

\[
5 + 30z + 7z = 3
\]
\[
30z + 7z = 3 - 5
\]
\[
37z = -2
\]
\[
z = -\frac{2}{37}
\]

### Step 5: Back substitute to find \( x \) and \( y \)

Now substitute \( z \) back into Equation 7 to find \( x \):

\[
x = 1 + 6\left(-\frac{2}{37}\right) = 1 - \frac{12}{37} = \frac{37}{37} - \frac{12}{37} = \frac{25}{37}
\]

Next, substitute \( z = -\frac{2}{37} \) into Equation 4 to find \( y \):

\[
y = 2 - \left(\frac{25}{37}\right) - 3\left(-\frac{2}{37}\right)
\]
\[
y = 2 - \frac{25}{37} + \frac{6}{37}
\]
\[
y = 2 - \frac{25 - 6}{37}
\]
\[
y = 2 - \frac{19}{37} = \frac{74}{37} - \frac{19}{37} = \frac{55}{37}
\]

### Final Solution

Thus, we find the solution:

\[
x = \frac{25}{37}, \quad y = \frac{55}{37}, \quad z = -\frac{2}{37}
\]

So the solution to the system of equations is:

\[
(x, y, z) = \left(\frac{25}{37}, \frac{55}{37}, -\frac{2}{37}\right)
\]