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To solve this problem, let's identify the given information and the formula we need.

### Given:
- \( M = (-4, 8) \) is the midpoint of line segment \( DE \).
- Point \( D \) has coordinates \( (6, 1) \).
- We need to find the coordinates of point \( E \).

### Midpoint Formula:
The midpoint \( M \) of a line segment connecting two points \( D(x_1, y_1) \) and \( E(x_2, y_2) \) is given by:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

### Apply the Midpoint Formula:
Since \( M = (-4, 8) \), \( D = (6, 1) \), and \( E = (x_2, y_2) \), we can set up equations to solve for \( x_2 \) and \( y_2 \).

1. For the x-coordinate:
\[
-4 = \frac{6 + x_2}{2}
\]

2. For the y-coordinate:
\[
8 = \frac{1 + y_2}{2}
\]

### Solve for \( x_2 \) and \( y_2 \):

1. Solve for \( x_2 \):
\[
-4 = \frac{6 + x_2}{2}
\]

Multiply both sides by 2:
\[
-8 = 6 + x_2
\]

Subtract 6 from both sides:
\[
x_2 = -8 - 6 = -14
\]

2. Solve for \( y_2 \):
\[
8 = \frac{1 + y_2}{2}
\]

Multiply both sides by 2:
\[
16 = 1 + y_2
\]

Subtract 1 from both sides:
\[
y_2 = 16 - 1 = 15
\]

### Coordinates of \( E \):
The coordinates of point \( E \) are \( (-14, 15) \).

Would you like further details or have any questions?

### Related Questions:
1. How would the solution change if \( D \) and \( M \) coordinates were different?
2. What if \( M \) were not the midpoint but rather some other ratio between \( D \) and \( E \)?
3. How can the distance formula be applied to verify the coordinates of \( E \)?
4. Can we find \( E \) if only the distance between \( D \) and \( E \) was given instead of the midpoint?
5. How does this method change in three-dimensional space?

### Tip:
When solving for coordinates, always check your work by substituting back into the original equations to ensure they are satisfied.

M(-4, 8) is the midpoint of DE. D has coordinates (6, 1). Find the coordinates of E.

Learn how to find the coordinates of point E when given the midpoint M(-4, 8) and another point D(6, 1) on the line segment DE. This problem covers key concepts in coordinate geometry, including the midpoint formula. Suitable for students in Grades 6-8.

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