The given parametric equations are:

\[
x = 8 + 2t, \quad y = 8 + 4t, \quad z = 5 + 6t
\]

These equations describe a straight line in 3D space, where \(t\) is the parameter.

### To better understand this:

1. The point at \(t = 0\) is \((x, y, z) = (8, 8, 5)\), which represents a specific point on the line.
2. The direction vector of the line is \(\vec{d} = (2, 4, 6)\), obtained from the coefficients of \(t\) in each equation.

This means the line passes through the point \((8, 8, 5)\) and is directed along the vector \((2, 4, 6)\). The parametric equations tell you the coordinates of any point on the line by varying \(t\).

Would you like further details on how to convert this to another form, or any specific aspect of this?

### Related Questions:
1. How do you convert parametric equations into symmetric form?
2. What is the interpretation of the direction vector?
3. How do you find the distance from a point to a line in 3D?
4. What is the equation of a line passing through two points?
5. How do you find the intersection of two lines in 3D?

### Tip:
The parameter \(t\) can represent time in physical problems, with the direction vector showing the velocity of motion along the line.

Explore how the parametric equations x = 8 + 2t, y = 8 + 4t, z = 5 + 6t describe a straight line in 3D space. Understand the direction vector (2, 4, 6) and how the line passes through the point (8, 8, 5). Learn about parametric equations, vector analysis, and their applications in 3D geometry, suitable for Grades 10-12.

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