Determine the relationship between x and y.:
\begin{tabular}{|c|c|}
\hline $\mathbf{x}$ & $\mathbf{y}$ \\
\hline 1 & 8 \\
\hline 2 & 12 \\
\hline 3 & 16 \\
\hline 4 & 20 \\
\hline
\end{tabular}

Step 1 ：Observe the given table and notice that as x increases by 1, y increases by 4. This suggests a linear relationship between x and y.

Step 2 ：Calculate the slope of the line that these points would form on a graph. The slope is the change in y divided by the change in x. Use any two points to calculate this. For example, use the points (1,8) and (2,12). The change in y is \(12-8=4\) and the change in x is \(2-1=1\). So the slope is \(4/1=4\). This means that for every increase of 1 in x, y increases by 4.

Step 3 ：Find the y-intercept by subtracting the product of the slope and the x-coordinate from the y-coordinate for any point. For example, use the point (1,8). The y-intercept is \(8 - 4*1 = 4\).

Step 4 ：Conclude that the relationship between x and y is given by the equation \(y = 4x + 4\).

Step 5 ：Final Answer: \(\boxed{y = 4x + 4}\)