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Let's break down the problem step-by-step to solve it. You are given two linear equations with constants \(a\) and \(b\):

\[
\begin{aligned}
    & x + ay + b = 0 \quad (1) \\
    & 3x - y - 2 = 0 \quad (2)
\end{aligned}
\]

### Analyzing the System of Equations:

1. **Equation (2)**: \( 3x - y - 2 = 0 \) can be rewritten as:
   \[
   y = 3x - 2
   \]

2. **Substitute \(y = 3x - 2\) into Equation (1)**:
   \[
   x + a(3x - 2) + b = 0
   \]
   Simplifying this:
   \[
   x + 3ax - 2a + b = 0
   \]
   Combine like terms:
   \[
   (1 + 3a)x + (-2a + b) = 0
   \]
   This is a linear equation in \(x\), so let's consider the cases depending on the values of \(a\) and \(b\):

### Now, let's address each statement:

#### 1. **The system has no solutions.**
   - The system has no solutions if the two lines are **parallel** but have different intercepts. 
   - From the equation above, if the coefficient of \(x\) is the same for both lines (parallel), this means \(1 + 3a = 0\), or \(a = -\frac{1}{3}\).
   - For the lines to not intersect, their constant terms must be different, i.e., \(-2a + b \neq 0\).
   - So, **sometimes true**. An example is \(a = -\frac{1}{3}\) and \(b \neq \frac{2}{3}\).

#### 2. **The system has precisely one solution.**
   - The system will have exactly one solution if the lines are **not parallel**, meaning they intersect at a single point.
   - For this to happen, \(1 + 3a \neq 0\), i.e., \(a \neq -\frac{1}{3}\).
   - So, **sometimes true**. An example is \(a = 1\) and \(b = 1\) (as given).

#### 3. **The system has precisely two solutions.**
   - A system of two linear equations cannot have exactly two solutions; it can either have one solution, no solution, or infinitely many solutions.
   - So, this is **never true**.

#### 4. **The system has infinitely many solutions.**
   - The system will have infinitely many solutions if the two equations represent the **same line**, which occurs when both the slopes and intercepts are identical.
   - This means \(1 + 3a = 0\) (or \(a = -\frac{1}{3}\)) and \(-2a + b = 0\).
   - So, **sometimes true**. An example is \(a = -\frac{1}{3}\) and \(b = \frac{2}{3}\).

### Final Answers:

1. **No solutions**: Sometimes true; example \(a = -\frac{1}{3}, b = 1\).
2. **One solution**: Sometimes true; example \(a = 1, b = 1\).
3. **Two solutions**: Never true.
4. **Infinitely many solutions**: Sometimes true; example \(a = -\frac{1}{3}, b = \frac{2}{3}\).

Would you like further clarification on any of the steps?

---

Here are 5 follow-up questions to expand on this topic:

1. How do you determine whether two lines are parallel?
2. What conditions are needed for two linear equations to have infinitely many solutions?
3. How can you geometrically interpret a system with no solutions?
4. Why can’t a system of linear equations have exactly two solutions?
5. Can a nonlinear system of equations have two solutions? How?

**Tip:** When solving a system of equations, it’s useful to check the slopes and intercepts to predict the number of solutions (none, one, or infinitely many).

You are given the two equations x + ay + b = 0 and 3x - y - 2 = 0, where a and b are constants. For each of the following statements, decide if it is always, sometimes, or never true. Provide examples for 'sometimes' cases.

Explore how to determine the number of solutions for a system of linear equations involving parameters. Understand when such systems have no solution, exactly one solution, or infinitely many solutions, with practical examples to illustrate each case.

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