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The given system of equations is:

\[
1. \quad x^2 + y^2 - 4x + 14y + 4 = 0
\]
\[
2. \quad x^2 + y^2 + 14x + 4y + 49 = 0
\]

To solve this system, we will attempt to rewrite each equation by completing the square for both \(x\) and \(y\).

### First equation: \( x^2 + y^2 - 4x + 14y + 4 = 0 \)

1. Group the \(x\) and \(y\) terms:
   \[
   (x^2 - 4x) + (y^2 + 14y) + 4 = 0
   \]
2. Complete the square for \(x\):
   \[
   x^2 - 4x = (x - 2)^2 - 4
   \]
3. Complete the square for \(y\):
   \[
   y^2 + 14y = (y + 7)^2 - 49
   \]
4. Substitute these into the equation:
   \[
   (x - 2)^2 - 4 + (y + 7)^2 - 49 + 4 = 0
   \]
5. Simplify:
   \[
   (x - 2)^2 + (y + 7)^2 - 49 = 0
   \]
   \[
   (x - 2)^2 + (y + 7)^2 = 49
   \]
This represents a circle with center \((2, -7)\) and radius \(7\).

### Second equation: \( x^2 + y^2 + 14x + 4y + 49 = 0 \)

1. Group the \(x\) and \(y\) terms:
   \[
   (x^2 + 14x) + (y^2 + 4y) + 49 = 0
   \]
2. Complete the square for \(x\):
   \[
   x^2 + 14x = (x + 7)^2 - 49
   \]
3. Complete the square for \(y\):
   \[
   y^2 + 4y = (y + 2)^2 - 4
   \]
4. Substitute these into the equation:
   \[
   (x + 7)^2 - 49 + (y + 2)^2 - 4 + 49 = 0
   \]
5. Simplify:
   \[
   (x + 7)^2 + (y + 2)^2 - 4 = 0
   \]
   \[
   (x + 7)^2 + (y + 2)^2 = 4
   \]
This represents a circle with center \((-7, -2)\) and radius \(2\).

### Interpretation:
The system of equations represents two circles:
1. A circle with center \((2, -7)\) and radius \(7\),
2. A circle with center \((-7, -2)\) and radius \(2\).

We can now solve for any intersections between these circles, but they describe two distinct circles in the plane. Would you like me to find the points of intersection between these circles or provide more details on their geometric properties?

### Follow-up questions:
1. How can we determine if two circles intersect geometrically?
2. What is the algebraic method to find the points of intersection between two circles?
3. What is the general formula for completing the square for any quadratic equation?
4. How can we describe the relationship between the radius of a circle and its area?
5. How would the equations change if we translated one of the circles?

### Tip:
When solving systems of equations involving circles, the distance between their centers relative to the sum of their radii can help determine if they intersect, are tangent, or are completely separate.

x^2 + y^2 - 4x + 14y + 4 = 0 and x^2 + y^2 + 14x + 4y + 49 = 0

Explore how to solve the system of circle equations x^2 + y^2 - 4x + 14y + 4 = 0 and x^2 + y^2 + 14x + 4y + 49 = 0. This problem involves completing the square to find the circle centers and radii, suitable for students in Grades 10-12.

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