https://file.aiquickdraw.com/mathbot/file/1727486779883wgp8fiey.jpg

To find the equation of the plane in the image, we need to utilize the information given:

- The plane passes through the point \((3, 1, 2)\).
- It is parallel to the \( xz \)-plane.

### Key observations:
1. A plane that is parallel to the \( xz \)-plane implies that the normal vector to the plane will be perpendicular to the \( xz \)-plane. Therefore, the normal vector will point in the direction of the \( y \)-axis.
2. Hence, the equation of such a plane will only involve the \( y \)-coordinate as a constant.

Since the plane is parallel to the \( xz \)-plane, the \( y \)-coordinate will remain fixed at \( y = 1 \), as the plane passes through the point \((3, 1, 2)\).

### Equation of the plane:
Thus, the equation of the plane is:
\[
y = 1
\]

This indicates that the plane lies at \( y = 1 \) for all \( x \) and \( z \)-values.

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Would you like more details on the reasoning or any further clarification? Here are some related questions you might find interesting:

1. What is the general equation of a plane given a point and a normal vector?
2. How do you determine the normal vector to a plane based on its orientation?
3. Can you explain how planes parallel to the coordinate planes are formed?
4. What changes when a plane is parallel to the \( xy \)- or \( yz \)-plane?
5. How can we find the distance between two parallel planes?

**Tip**: When a plane is parallel to a coordinate plane, its equation simplifies by excluding the variable in the direction perpendicular to that coordinate plane.

The plane below goes through the point (3, 1, 2) and it is parallel to the xz-plane. What is the equation of the plane shown?

Find the equation of a plane that passes through the point (3, 1, 2) and is parallel to the xz-plane. This problem involves understanding the properties of planes in 3D geometry and how to derive the equation of a plane based on its orientation.

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