How to find the parametric equations for the line through the point that is parallel to the plane.

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How to find the equations of the line passing through the point $(8, 7, 9)$, intersecting the $z$-axis and parallel to the plane $2x−6y+z−12=0$.

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Hint: Let $A(8,7,9)$ and $M(0,0,z)$ the intersection of the sought line with the $z$. If you determine $M$, the parametric equation will follow.

The line $AM$ is parallel to the plane if and only if $\vec{AM}$ is orthogonal to the normal vector of the plane $\vec n\,(2,-6,1)$.