In this problem, you are given the image of point $P(x, y, z)$ and its image $P'(1, 0, 7)$ with respect to a line. The equation of the line is given as:

$\frac{x + 1}{2} = \frac{y - 1}{3} = \frac{z - 2}{5}$

You are tasked with finding the coordinates of point $P$.

Approach:

To solve this, you can use the fact that $P'$ is the image of $P$ reflected across the line. The midpoint $M$ of $P$ and $P'$ lies on the line. The coordinates of $M$ can be expressed as a function of the parameter $\lambda$ that defines the line:

$M = (\lambda, 2\lambda + 1, 3\lambda + 2)$

Given that the coordinates of $P'$ are $(1, 0, 7)$, the distances along the x, y, and z coordinates are:

$\frac{x + 1}{2} = 1, \quad \frac{y + 0}{2} = 3, \quad \frac{z + 7}{2} = 5$

By solving these equations, we can find the coordinates of point $P$. The detailed steps in your provided image give the solution as $P(1, 6, 3)$.

Would you like more details on any step? Or do you have any other questions related to this problem?

Follow-up Questions:

1. How do you calculate the midpoint of two points in 3D geometry?
2. What is the general formula for the reflection of a point across a line?
3. How do you solve the system of linear equations involving the parameters of a line?
4. What is the geometric interpretation of the equation for the line in the problem?
5. How can you use the concept of parametric equations to solve problems in 3D geometry?

Tip: When dealing with reflections and midpoints in 3D geometry, always check if the midpoint lies on the line or the plane where the reflection occurs.