Example 1. Find the equation of the line passing through the intersection points of the two curves
$$
3 x^{2}+3 y^{2}+2 x+y=0 \text { and } x^{2}+y^{2}+3 x
$$
$-\mathrm{y}=0$.
Solution 1 Solve the system of equations
$$
\left\{\begin{array}{l}
x^{2}+y^{2}+3 x-y=0 \\
3 x^{2}+3 y^{2}+2 x+y=0
\end{array}\right.
$$

Find the two intersection points, then use the two-point form to find the equation of the line.
Solution 2 The equation $\left(3 x^{2}+3 y^{2}+2 x+y\right)$ $+\lambda\left(x^{2}+y^{2}+3 x-y\right)=0$ represents the curve passing through the intersection points of these two curves for any real number $\lambda$. Only when $\lambda=-3$, can the quadratic terms be eliminated. At this point, the equation simplifies to $7 x-4 y=0$, which is the desired equation.