Example 5. Find the equation of the line containing the common chord of two intersecting circles: $x^{2}+y^{2}+D_{1} x+E_{1} y$ $+F_{1}=0, x^{2}+y^{2}+D_{2} x+E_{2} y+F_{2}=0$.
Let the common points of the two circles be \( C\left(x_{1}, y_{1}\right) \) and \( D\left(x_{2}, y_{2}\right) \). Then \( C\left(x_{1}, y_{1}\right) \) should satisfy the equations of both circles:
\[
\begin{array}{c}
x_{1}^{2} + y_{1}^{2} + D_{1} x_{1} + E_{1} y_{1} + F_{1} = 0, \\
\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\
x_{1}^{2} + y_{1}^{2} + D_{2} x_{1} + E_{2} y_{1} + F_{2} = 0 .
\end{array}
\]
(1) - (2) gives
\[
\begin{aligned}
& \left(D_{1} - D_{2}\right) x_{1} + \left(E_{1} - E_{2}\right) y_{1} + F_{1} - F_{2} \\
= & 0 .
\end{aligned}
\]

This equation shows that \( \left(x_{1}, y_{1}\right) \) is a solution to the linear equation \( \left(D_{1} - D_{2}\right) x + \left(E_{1} - E_{2}\right) y + F_{1} - F_{2} = 0 \). By condition \( I \), we know: point \( C\left(x_{1}, y_{1}\right) \) lies on the line \( \left(D_{1} - D_{2}\right) x + \left(E_{1} - E_{2}\right) y + F_{1} - F_{2} = 0 \).
Similarly, we can see that point \( D\left(x_{2}, y_{2}\right) \) also lies on this line.
Since two points determine a line, the equation of the line containing the common chord of the two circles is
\[
\left(D_{1} - D_{2}\right) x + \left(E_{1} - E_{2}\right) y + F_{1} - F_{2} = 0
\]