Example 1.21. Form the equation of the plane passing through the line of intersection of the planes $x+y+5z-1=0$, $2x+3y-z+2=0$ and through the point $M(3,2,1)$.
Solution. We will use the equation of a pencil of planes

$$
x+y+5 z-1+\lambda(2 x+3 y-z+2)=0 .
$$

The value of $\lambda$ is determined from the condition that the coordinates of point $M$ must satisfy this equation:

$$
\begin{gathered}
3+2+5-1+\lambda(6+6-1+2)=9+13 \lambda=0 \\
\lambda=-\frac{9}{13}
\end{gathered}
$$

We obtain the desired equation in the form:

$$
x+y+5 z-1-\frac{9}{13}(2 x+3 y-z+2)=0
$$

or, multiplying by 13 and combining like terms, in the form:

$$
5 x+14 y-74 z+31=0
$$