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The question is asking to select all the lines that:

1. Contain the point \((5, 7, 3)\).
2. Are parallel to the vector \(\langle 1, 2, 3 \rangle\).

For parametric equations, a line in 3D can be written in the form:
\[
x = x_0 + at, \quad y = y_0 + bt, \quad z = z_0 + ct
\]
Where \((x_0, y_0, z_0)\) is a point on the line and \(\langle a, b, c \rangle\) is the direction vector of the line.

### Step-by-step check:
- The direction vector of the line must be \(\langle 1, 2, 3 \rangle\).
- The point \((5, 7, 3)\) must satisfy the line's equation when \(t = 0\).

### Now, let's analyze each equation set:

1. **\(x = 6 + 2t, y = 4 + 4t, z = -8 + 6t\)**:
   - Direction vector: \(\langle 2, 4, 6 \rangle\), which simplifies to \(\langle 1, 2, 3 \rangle\) when divided by 2. This line has the correct direction.
   - Check if the point \((5, 7, 3)\) lies on the line:
     - At \(t = 0\): \(x = 6\), \(y = 4\), \(z = -8\). This does **not** contain the point \((5, 7, 3)\).
   - **Not a valid line.**

2. **\(x = 5 + t, y = 7 + 2t, z = 3 + 3t\)**:
   - Direction vector: \(\langle 1, 2, 3 \rangle\). This matches the given direction.
   - At \(t = 0\): \(x = 5\), \(y = 7\), \(z = 3\), so the point \((5, 7, 3)\) lies on this line.
   - **Valid line.**

3. **\(x = 3 + t, y = 3 + 2t, z = -3 + 3t\)**:
   - Direction vector: \(\langle 1, 2, 3 \rangle\). This matches the given direction.
   - At \(t = 2\): \(x = 5\), \(y = 7\), \(z = 3\), so the point \((5, 7, 3)\) lies on this line.
   - **Valid line.**

4. **\(x = 4 + t, y = 5 + 2t, z = 3 + 3t\)**:
   - Direction vector: \(\langle 1, 2, 3 \rangle\). This matches the given direction.
   - At \(t = 1\): \(x = 5\), \(y = 7\), \(z = 3\), so the point \((5, 7, 3)\) lies on this line.
   - **Valid line.**

### Conclusion:
The valid lines are:
- \(x = 5 + t, y = 7 + 2t, z = 3 + 3t\)
- \(x = 3 + t, y = 3 + 2t, z = -3 + 3t\)
- \(x = 4 + t, y = 5 + 2t, z = 3 + 3t\)

Do you need further details or have any questions?

### Related Questions:
1. What is the general form of a line equation in 3D?
2. How do you find if two vectors are parallel?
3. How can parametric equations be derived from a vector equation?
4. What does it mean for two vectors to be proportional?
5. How can the position of a point on a line be calculated using parametric equations?

**Tip**: Always check if a given point satisfies all parametric equations for a line to ensure it's part of the line.

Select ALL the lines that contain the point (5, 7, 3) and are parallel to the vector ⟨1, 2, 3⟩.

Learn how to determine which lines in parametric form contain the point (5, 7, 3) and are parallel to the vector ⟨1, 2, 3⟩. This problem involves parametric equations and vector geometry in 3D space, and is suitable for advanced high school or undergraduate students.

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