Find the equation of the line that passes through the centers of the circle C1: x raised 2 + y raised 2 = 4 and C2: x raised 2 + y raised 2 + 6x -2y = 0

1. Recognize the center of the first circle \(C1: x^2 + y^2 = 4\):

(h, k) = (0, 0)

2. Rewrite the equation of the second circle \(C2: x^2 + y^2 + 6x - 2y = 0\) in its standard form:

(x + 3)^2 + (y - 1)^2 = 10

3. Identify the center of the second circle:

(h, k) = (-3, 1)

4. Calculate the slope \(m\) of the line passing through the centers \((0, 0)\) and \((-3, 1)\):

m = \frac{1 - 0}{-3 - 0} = -\frac{1}{3}

5. Use the point-slope form to write the equation of the line:

y - 0 = -\frac{1}{3}(x - 0)

6. Simplify to get the final equation:

y = -\frac{1}{3}x