To solve the given system of equations, we first analyze the dependencies between the equations and then use elimination/substitution to find the general solution.

Step 1: Identify Dependent Equations

The system is:

1. (x + 2y + z = 1)
2. (2x + 3y + z = 3)
3. (3x + 5y + 2z = 4)

Notice that Equation 3 = Equation1 + Equation2 (both sides: (1+3=4)), so Equation3 is dependent. We only have 2 independent equations for 3 variables, meaning infinitely many solutions.

Step 2: Parametrize the Solution

Let (z = t) (parameter, (t \in \mathbb{R})).

From Equation2 - Equation1:
(x + y = 2) → (x = 2 - y)

Substitute (x = 2 - y) and (z = t) into Equation1:
((2 - y) + 2y + t = 1) → (2 + y + t =1) → (y = -1 -t)

Then (x = 2 - (-1 -t) = 3 + t)

General Solution

(x = 3 + t), (y = -1 -t), (z = t) (for any real (t))

Example solution: When (t=0), (x=3), (y=-1), (z=0).

Answer: Infinitely many solutions, e.g., (x=3), (y=-1), (z=0) (or the parametric form (x=3+t), (y=-1-t), (z=t)).

For a specific solution, (\boxed{x=3, y=-1, z=0}) (taking (t=0)).

(\boxed{x=3, y=-1, z=0})

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