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**Hint:**Change of form of the given equation $y-2=\dfrac{2}{3}x$ will give the slope of the line. We change it to the form of $y=mx+k$ to find the slope m. Then, we get into the form of its parallel line to find the equation. We put the point value of $\left( 3,2 \right)$ to find the value of the constant. We get the equation of the parallel line.

**Complete step by step answer:**

The given equation of the line $y-2=\dfrac{2}{3}x$ can be changed to $y=\dfrac{2}{3}x+2$.

The given equation $y=\dfrac{2}{3}x+2$ is of the form $y=mx+k$. m is the slope of the line.

This gives that the slope of the line $y-2=\dfrac{2}{3}x$ is $\dfrac{2}{3}$.

We know that any line parallel to the given line will be of the same slope.

This means any line parallel to $y-2=\dfrac{2}{3}x$ will have a slope of $\dfrac{2}{3}$.

We take the equation of any parallel line to $y=mx+k$ as $y=mx+c$.

Let’s assume that the parallel line to $y=\dfrac{2}{3}x+2$ as $y=\dfrac{2}{3}x+p$. Here $p$ is a constant value which we have to find out.

The parallel line goes through the point $\left( 3,2 \right)$. We place the point in the equation $y=\dfrac{2}{3}x+p$.

$\begin{align}

& 2=\dfrac{2}{3}\times 3+p \\

& \Rightarrow p=2-2=0 \\

\end{align}$

Value of $p$ is 0.

The equation of the line is $y=\dfrac{2}{3}x$. Simplified form is $3y=2x$.

**Note:**A line parallel to the X-axis does not intersect the X-axis at any finite distance and hence we cannot get any finite x-intercept of such a line. Same goes for lines parallel to the Y-axis. In case of slope of a line the range of the slope is 0 to $\infty $.

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