To solve these equations for \(x\) and \(y\), you'll need to manipulate each one individually. I'll guide you on how to approach each equation:

1. **Equation 1:**
\[
x^2 + y^2 + 4y + 13 = 6x
\]
- Rearrange terms:
\[
x^2 - 6x + y^2 + 4y + 13 = 0
\]
- Complete the square for \(x\) and \(y\):
\[
(x-3)^2 - 9 + (y+2)^2 - 4 + 13 = 0
\]
\[
(x-3)^2 + (y+2)^2 = 0
\]
- Solve the simplified equation: This implies \(x = 3\) and \(y = -2\).

2. **Equation 2:**
\[
x^2 + y^2 + 17 = 2x - 8y
\]
- Rearrange terms:
\[
x^2 - 2x + y^2 + 8y + 17 = 0
\]
- Complete the square:
\[
(x-1)^2 - 1 + (y+4)^2 - 16 + 17 = 0
\]
\[
(x-1)^2 + (y+4)^2 = 0
\]
- Solve the simplified equation: This implies \(x = 1\) and \(y = -4\).

3. **Equation 3:**
\[
x^2 + y^2 + 45 = 12y - 6x
\]
- Rearrange terms:
\[
x^2 + 6x + y^2 - 12y + 45 = 0
\]
- Complete the square:
\[
(x+3)^2 - 9 + (y-6)^2 - 36 + 45 = 0
\]
\[
(x+3)^2 + (y-6)^2 = 0
\]
- Solve the simplified equation: This implies \(x = -3\) and \(y = 6\).

4. **Equation 4:**
\[
4x^2 + 9y^2 + 2 = 4x + 6y
\]
- Rearrange terms:
\[
4x^2 - 4x + 9y^2 - 6y + 2 = 0
\]
- Complete the square:
\[
4(x-\frac{1}{2})^2 - 1 + 9(y-\frac{1}{3})^2 - \frac{2}{3} + 2 = 0
\]
\[
4(x-\frac{1}{2})^2 + 9(y-\frac{1}{3})^2 = \frac{5}{3}
\]
- \(x = \frac{1}{2}\) and \(y = \frac{1}{3}\) can solve the equation under these constraints.

Verify these solutions by substituting back into the original equations.