The problem is asking which relationships have the same constant of proportionality between $y$ and $x$ as the equation $y = \frac{5}{2} x$.

To solve this, we will analyze each option and check if the constant of proportionality (the ratio $\frac{y}{x}$) is equal to $\frac{5}{2}$.

1. Option A: $5y = 2x$

   • Rearrange to get $y$ in terms of $x$: $y = \frac{2}{5} x$
   • The constant of proportionality here is $\frac{2}{5}$, which is not equal to $\frac{5}{2}$.

2. Option B: $8y = 20x$

   • Rearrange to get $y$ in terms of $x$: $y = \frac{20}{8} x = \frac{5}{2} x$
   • The constant of proportionality is $\frac{5}{2}$, which matches the given equation.

3. Option C: Graph with a line passing through points (0,0) and (2,5)

   • The slope (constant of proportionality) is given by: $\text{slope} = \frac{\Delta y}{\Delta x} = \frac{5 - 0}{2 - 0} = \frac{5}{2}$
   • The constant of proportionality is $\frac{5}{2}$, which matches the given equation.

4. Option D: Graph with a line passing through points (0,0) and (5,2)

   • The slope is: $\text{slope} = \frac{\Delta y}{\Delta x} = \frac{2 - 0}{5 - 0} = \frac{2}{5}$
   • The constant of proportionality is $\frac{2}{5}$, which is not equal to $\frac{5}{2}$.

5. Option E: Table of values

   • To check the proportionality, calculate $\frac{y}{x}$ for each pair of values:
     • For $x = 1$, $y = 2$, we have $\frac{y}{x} = \frac{2}{1} = 2$.
     • For $x = 4$, $y = 10$, we have $\frac{y}{x} = \frac{10}{4} = \frac{5}{2}$.
     • For $x = 7$, $y = 17$, we have $\frac{y}{x} = \frac{17}{7}$, which is not $\frac{5}{2}$.
   • Since not all pairs of values match $\frac{5}{2}$, this option does not match.

Final answer:

The relationships that match the given equation are:

• B
• C

Would you like more details on how the slopes were calculated?

Here are 5 related questions:

1. What does a constant of proportionality represent in a linear relationship?
2. How can you determine the constant of proportionality from a graph?
3. How would you convert an equation into slope-intercept form to find the constant of proportionality?
4. How does the constant of proportionality change with different line slopes?
5. What are the characteristics of direct proportionality in algebraic expressions?

Tip: In equations, always isolate $y$ to find the constant of proportionality (slope), which makes comparisons easier!