To find the equation of a plane that is parallel to a given plane and passes through a specific point, we can follow these steps:
1. **Understand the Plane Equation**: The equation of the given plane is in the standard form Ax + By + Cz + D = 0, where the coefficients (A, B, C) represent the normal vector to the plane. For the plane 2x + y – 3z – 7 = 0, the normal vector is (2, 1, -3).
2. **Use the Normal Vector**: Since we need a plane that is parallel to this plane, it must have the same normal vector. Therefore, the equation of the new plane will also have the form 2x + y – 3z + D = 0.
3. **Plug in the Point**: Now we need to find the constant D using the point we’re given, which is (1, 4, 2). We substitute x = 1, y = 4, and z = 2 into the equation:
2(1) + 4 – 3(2) + D = 0
Simplifying this, we have:
2 + 4 – 6 + D = 0
0 + D = 0
So, D = 0.
4. **Write the Final Equation**: Therefore, the equation of the new plane that passes through the point (1, 4, 2) and is parallel to the given plane is:
2x + y – 3z = 0.