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.