Relative position of line and plane
A distinction is made between three possible relative positions between a line $g$ and a plane $E$.
 $g$ and $E$ intersect
 $g$ and $E$ are parallel
 $g$ lies in plane $E$
Remember
 clear intersection: $g$ and $E$ intersect
(an intersection)  wrong statement (e.g. $0=5$): $g$ parallel to $E$
(no intersection)  true statement (z. B. $5=5$): $g$ is in $E$
(infinite intersections)
Hint
If the plane is in the parametric form, one would have to solve a system of linear equations with three equations and variables, which should be avoided due to the complexity.
Method
 Rewrite equation of a line
 Insert $x$, $y$, $z$ in the plane's cartesian equation and solve
 Interpret the result
Example
$\text{g: } \vec{x} = \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix} + r \cdot \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}$
$\text{E: } 2x+y+2z=2$

Rewrite equation of a line
The vector $\vec{x}$ in the equation of a line is replaced by $\begin{pmatrix} x \\ y \\ z \end{pmatrix}$.$\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix} + r \cdot \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}$
Each line corresponds to an equation
 $x=\color{red}{2+2r}$
 $y=\color{blue}{13r}$
 $z=\color{green}{1+4r}$

Insert $x$, $y$, $z$
The individual equations for $x$, $y$, $z$ can be inserted into the plane's cartesian equation.$\text{E: } 2\color{red}{x}+\color{blue}{y}+2\color{green}{z}=2$
$2\cdot\color{red}{(2+2r)}$ $+\color{blue}{(13r)}$ $+2\cdot\color{green}{(1+4r)}$ $=2$
Now the brackets are solved and the equation is converted to $r$
$4+4r+13r+2+8r$ $=2$
$7+9r=2\quad7$
$9r=9\quad:9$
$r=1$ 
Interpret the result
Since we have got a clear $r$, the plane and the straight line must intersect and you can calculate the intersection.=> Line $g$ and plane $E$ intersect.
The intersection is calculated by inserting $r=1$ into the equation of a line.
$\begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix} + (1) \cdot \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}$ $=\begin{pmatrix} 0 \\ 4 \\ 3 \end{pmatrix}$
=> intersection $S(043)$.