The distance between the parallel planes
10x + 2y - 2z = 5 and 5x + y - z = 1 is equal to

To find the distance between two parallel planes, we can use the formula:

\[
\text{Distance} = \frac{|d_2 - d_1|}{\sqrt{A^2 + B^2 + C^2}}
\]

where \(Ax + By + Cz = d_1\) and \(Ax + By + Cz = d_2\) are the equations of the planes.

### Step 1: Rewrite the equations of the planes in the standard form.

1. **First plane:** \(10x + 2y - 2z = 5\)

   We can write this as:
   \[
   10x + 2y - 2z - 5 = 0 \quad \text{(here, } d_1 = 5\text{)}
   \]

2. **Second plane:** \(5x + y - z = 1\)

   We can write this as:
   \[
   5x + y - z - 1 = 0 \quad \text{(here, } d_2 = 1\text{)}
   \]

### Step 2: Normalize the second plane to have the same coefficients as the first.

To express the second plane in the form compatible with the first, we can multiply the equation of the second plane by 2:

\[
2(5x + y - z) = 2(1) \implies 10x + 2y - 2z = 2
\]

Now, we have:
- First plane: \(10x + 2y - 2z = 5\) (where \(d_1 = 5\))
- Second plane (normalized): \(10x + 2y - 2z = 2\) (where \(d_2 = 2\))

### Step 3: Calculate the distance.

Substituting \(d_1 = 5\) and \(d_2 = 2\) into the distance formula:

\[
\text{Distance} = \frac{|2 - 5|}{\sqrt{10^2 + 2^2 + (-2)^2}} = \frac{|-3|}{\sqrt{100 + 4 + 4}} = \frac{3}{\sqrt{108}} = \frac{3}{6\sqrt{3}} = \frac{1}{2\sqrt{3}}
\]

### Final Answer

Thus, the distance between the two parallel planes is:

\[
\frac{1}{2\sqrt{3}} \text{ or approximately } 0.2887.
\]

If you would like more details or have any questions, let me know! Here are some related questions:

1. What are the general equations of parallel planes?
2. How do you determine if two planes are parallel?
3. Can you provide an example of finding the distance between non-parallel planes?
4. What is the geometric interpretation of the distance between planes?
5. How do you calculate the angle between two intersecting planes?

**Tip:** When finding distances, always ensure the planes are expressed in a consistent form before applying formulas.

The distance between the parallel planes 10x + 2y - 2z = 5 and 5x + y - z = 1 is equal to

Discover how to calculate the distance between the parallel planes 10x + 2y - 2z = 5 and 5x + y - z = 1 using the distance formula for planes. This detailed guide is suitable for students in Grades 10-12.

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