$$ \begin{array}{|r|r|} \hline x & y \\ \hline-3 b & 18 b \\ \hline-2 b & 13 b \\ \hline 0 & 3 b \\ \hline 2 b & -7 b \\ \hline \end{array} $$ In the table above, \(b\) is a constant. If the \(x y\)-table describes some points on a linear function between \(x\) and \(y\), which of the following equations could represent that function? A) \(5 x+y=2 b\) B) \(x-5 y=-3 b\) C) \(5 x+y=3 b\) D) \(x-5 y=-7 b\)

Let's consider two points from the table to find the slope (rate of change). We can use the points \((-3b, 18b)\) and \((-2b, 13b)\).

Using the two points chosen in step 1, we can calculate the slope (rate of change) as follows: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) \\ \(m = \frac{13b - 18b}{-2b - (-3b)}\) \\ \(m = \frac{-5b}{b}\) \\ \(m = -5\)

Now we will test the given options to see if they have the same slope as we calculated: A) \(5x + y = 2b\) The slope of this equation is \(-5\) (adjust to the slope-intercept form: \(y = -5x + 2b\)) which matches our calculated slope. However, let's quickly verify if the other options are incorrect before confirming our choice: B) \(x - 5y = -3b\) The slope of this equation is \(\frac{1}{5}\) (adjust to the slope-intercept form: \(y = \frac{1}{5}x + \frac{3}{5}b\)) which doesn't match our calculated slope. C) \(5x + y = 3b\) The slope of this equation is \(-5\) (adjust to the slope-intercept form: \(y = -5x + 3b\)) which matches our calculated slope. D) \(x - 5y = -7b\) The slope of this equation is \(\frac{1}{5}\) (adjust to the slope-intercept form: \(y = \frac{1}{5}x + \frac{7}{5}b\)) which doesn't match our calculated slope. Now we have two options left: A and C. To find out which one is correct, we need to plug in an \((x,y)\) pair from the table into both equations and see which one produces a true statement.

Let's use the point \((0, 3b)\) from the table to test our remaining options: A) \(5x + y = 2b\) Plugging in \((0, 3b)\): \( 5(0) + 3b = 2b\) No, this equation doesn't satisfy with this point. So let's try C. C) \(5x + y = 3b\) Plugging in \((0, 3b)\): \( 5(0) + 3b = 3b\) Yes, this equation satisfies with this point. So, the correct equation that represents the linear function in the given table is: C) \(5x + y = 3b\)