Find the equation of the plane through the point P = (5, 5, 5) and parallel to the plane 5x + 5y + 4z = −6.

To find the equation of a plane that passes through a given point \(P = (5, 5, 5)\) and is parallel to a given plane \(5x + 5y + 4z = -6\), we can use the fact that parallel planes have the same normal vector.

The given plane \(5x + 5y + 4z = -6\) has a normal vector \(\mathbf{n} = \langle 5, 5, 4 \rangle\).

Since the new plane is parallel to the given plane, it will have the same normal vector. The general equation of a plane with normal vector \(\langle A, B, C \rangle\) is:

\[Ax + By + Cz = D\]

Using the normal vector \(\mathbf{n} = \langle 5, 5, 4 \rangle\), the equation of our new plane is:

\[5x + 5y + 4z = D\]

To find \(D\), we use the point \(P = (5, 5, 5)\) which lies on the plane. Substituting \(P\) into the plane equation:

\[5(5) + 5(5) + 4(5) = D\]

Calculating the left side:

\[25 + 25 + 20 = 70\]

So, \(D = 70\). Therefore, the equation of the plane passing through \(P = (5, 5, 5)\) and parallel to the plane \(5x + 5y + 4z = -6\) is:

\[5x + 5y + 4z = 70\]

Would you like further details or have any questions?

Here are 8 related questions that expand on this topic:

1. How do you determine the normal vector of a plane given its equation?
2. What is the general form of a plane equation in three dimensions?
3. How do you find the distance between a point and a plane?
4. What is the geometric significance of the normal vector of a plane?
5. How can you find the intersection line of two planes?
6. How do you determine if two planes are perpendicular?
7. How do you find the angle between a line and a plane?
8. How can you find the equation of a plane given three points on the plane?

**Tip:** When writing the equation of a plane, always ensure that the coefficients of \(x\), \(y\), and \(z\) represent the components of the normal vector to the plane.

Learn how to find the equation of a plane passing through a given point and parallel to a specified plane in 3D space. This problem involves vectors and the equation of a plane, with step-by-step calculations and examples. Suitable for advanced high school students and anyone interested in 3D geometry problems.

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