The distance between the parallel planes 10x + 2y + 22 = 10 and 5x+y+z= 2 is

To find the distance between two parallel planes, we first need to express both planes in the standard form \(Ax + By + Cz + D = 0\).

1. **First Plane**: 
   \[
   10x + 2y + 22 = 10 \implies 10x + 2y + 12 = 0
   \]

2. **Second Plane**: 
   \[
   5x + y + z = 2 \implies 5x + y + z - 2 = 0
   \]

To determine if these planes are parallel, we compare their normal vectors:
- For the first plane, the normal vector is \(\vec{n_1} = (10, 2, 0)\).
- For the second plane, the normal vector is \(\vec{n_2} = (5, 1, 1)\).

Since the normal vectors are not scalar multiples of each other, the planes are not parallel. Therefore, we cannot calculate the distance between them as if they were parallel.

### Summary:
The planes \(10x + 2y + 22 = 10\) and \(5x + y + z = 2\) are not parallel. 

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### Additional Questions:
1. Would you like to know how to determine the angle between two planes?
2. Do you want to find the distance from a point to one of the planes?
3. Are you interested in how to derive the equations of a plane from given points?
4. Would you like to learn more about the geometric interpretation of planes in space?
5. Do you want to solve a different problem related to planes or distances?

### Tip:
To verify if two planes are parallel, check if their normal vectors are proportional.

The distance between the parallel planes 10x + 2y + 22 = 10 and 5x + y + z = 2 is

Discover why the planes 10x + 2y + 22 = 10 and 5x + y + z = 2 are not parallel and learn how to determine distances in geometry. This detailed solution explains the concepts of normal vectors and provides insights into plane equations. Ideal for high school students in Grades 10-12.

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