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The task is to identify which parametric equations for lines pass through the point \( (5, 7, 3) \) and are parallel to the vector \( \langle 4, 2, 3 \rangle \).

Let's break this down:

1. **Condition 1: The lines must contain the point \( (5, 7, 3) \):**
   - For each line, check if setting \( t = 0 \) gives the coordinates \( (5, 7, 3) \).

2. **Condition 2: The lines must be parallel to the vector \( \langle 4, 2, 3 \rangle \):**
   - In the parametric form of a line \( x = x_0 + a t \), \( y = y_0 + b t \), \( z = z_0 + c t \), the vector \( \langle a, b, c \rangle \) must match the direction vector \( \langle 4, 2, 3 \rangle \).

Now, let's analyze each option:

### First Equation:
\[
x = 6 + 2t, \quad y = 4 + 4t, \quad z = 8 + 6t
\]
- Direction vector: \( \langle 2, 4, 6 \rangle \). This is not parallel to \( \langle 4, 2, 3 \rangle \), since it is not a scalar multiple of \( \langle 4, 2, 3 \rangle \).
- Therefore, this line is not parallel to the given vector.

### Second Equation:
\[
x = 5 + t, \quad y = 7 + 2t, \quad z = 3 + 3t
\]
- Direction vector: \( \langle 1, 2, 3 \rangle \). This is also not parallel to \( \langle 4, 2, 3 \rangle \), as the \( x \)-component differs.
- Hence, this line is not parallel to the given vector.

### Third Equation:
\[
x = 3 + 4t, \quad y = 3 + 2t, \quad z = -3 + 3t
\]
- Direction vector: \( \langle 4, 2, 3 \rangle \), which is exactly the direction vector we're looking for.
- Check if the line contains the point \( (5, 7, 3) \). Set \( t \) such that:
  - \( 3 + 4t = 5 \) → \( t = \frac{1}{2} \)
  - \( 3 + 2t = 7 \) → \( t = 2 \) (Contradiction)
- Therefore, this line does **not** pass through the point \( (5, 7, 3) \).

### Fourth Equation:
\[
x = 4 + t, \quad y = 5 + 2t, \quad z = -3t
\]
- Direction vector: \( \langle 1, 2, -3 \rangle \). This does not match \( \langle 4, 2, 3 \rangle \).
- Hence, this line is not parallel to the given vector.

### Conclusion:
None of the provided lines satisfy **both** conditions: passing through the point \( (5, 7, 3) \) and being parallel to the vector \( \langle 4, 2, 3 \rangle \).

Would you like a more detailed explanation on any part, or have further questions?

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**Related Questions:**

1. How do you determine if two vectors are parallel?
2. What is the parametric form of a line in 3D space?
3. How can you find if a point lies on a given line?
4. What does it mean for two lines to be parallel in 3D space?
5. How do you find the equation of a line given a point and a direction vector?

**Tip:** When checking if a line passes through a point in parametric form, always substitute \( t = 0 \) to check if the starting point matches.

Select all the lines that contain the point (5, 7, 3) and are parallel to the vector (4, 2, 3).

Analyze parametric equations to find lines that pass through the point (5, 7, 3) and are parallel to the vector (4, 2, 3). Learn how to check the conditions for parallelism and containment using vector properties and parametric equations.

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