Answer
$$3\lambda x+by+2 \lambda z=5\lambda.\\
\lambda\neq 0.$$
Work Step by Step
The equation of any plane that intersects the $ xz $-plane in the line $3x+2z=5$ is given by
$$ ax+by+cz=d.$$
Now, to find the $ xz $ intersection, we put $ y=0$ and hence we get
$$ ax+ cz=d.$$
Due to the intersection, we have
$$ a=3\lambda, \quad c=2\lambda, \quad d=5\lambda, \quad \lambda\neq 0.$$
So the equation of any plane whose intersection is the line $3x+2z=5$ is given by
$$3\lambda x+by+2 \lambda z=5\lambda.$$
