Example 1.22. Form the equation of the plane passing through the line of intersection of the planes $x+3 y+5 z-4=0$ and $x-y-2 z+7=0$ and parallel to the y-axis.
Solution. We will use the equation of the pencil $x+3 y+5 z-4+$ $+\lambda(x-y-2 z+7)=0, \quad$ transform the equation to the form $(1+\lambda) x+(3-\lambda) y+(5-2 \lambda) z+(7 \lambda-4)=0$.

Since the desired plane is parallel to the y-axis, the coefficient of y should be zero, i.e., $3-\lambda=0, \lambda=3$. Substituting the value of $\lambda$ into the equation of the pencil, we get

$$
4 x-z+17=0 .
$$