To find the linear equation that describes the relationship between \(x\) and \(y\) based on the table provided, we first identify the pattern.
Looking at the table:
\[ \begin{array}{c|c} x & y \ \hline 5 & 47 \ 6 & 49 \ 7 & 51 \ 8 & 53 \ \end{array} \]
We can see that as \(x\) increases by 1, \(y\) increases by 2. This indicates that the slope \(m\) of the line is 2.
Now we can use the point-slope form of the equation of a line, or start from the slope-intercept form \(y = mx + b\).
Given the slope \(m = 2\), we can now substitute in a point from the table to find \(b\). Let's use the point (5, 47):
\[ y = 2x + b \]
Substituting \(x = 5\) and \(y = 47\):
\[ 47 = 2(5) + b \] \[ 47 = 10 + b \] \[ b = 47 - 10 = 37 \]
Now, we can write the linear equation:
\[ y = 2x + 37 \]
Thus, the final answer is:
\[ y = 2x + 37 \]