`{:(x,2,4,6,8,10),(y,7/5,11/5,15/5,19/5,23/5):}`
Which of the following equations realates y to x accoreing to the values shown in the table above ?

Options

A. `y= ((2)/(5))^(x) -7/5`
B. `y = ((3x)/(5))^(2) -2`
C. `y = 5/2x- 3/5`
D. `y = 2/5 x + 3/5`

Explanation

To find the equation that relates \( y \) to \( x \) based on the values given in the table, we will follow these steps:

### Step 1: Identify the pattern in the table

The table provides the following values:

\[ \begin{array}{|c|c|} \hline x & y \\ \hline 2 & \frac{7}{5} \\ 4 & \frac{11}{5} \\ 6 & \frac{15}{5} \\ 8 & \frac{19}{5} \\ 10 & \frac{23}{5} \\ \hline \end{array} \]

From the table, we can observe that as \( x \) increases by 2, \( y \) increases by \( \frac{4}{5} \).

### Step 2: Determine the slope of the linear relationship

The change in \( y \) for a change in \( x \) can be calculated as follows:

\[ \text{Slope} = \frac{\Delta y}{\Delta x} = \frac{\frac{4}{5}}{2} = \frac{2}{5} \]

This indicates that for every increase of 2 in \( x \), \( y \) increases by \( \frac{4}{5} \).

### Step 3: Find the y-intercept

To find the y-intercept, we can use one of the points from the table. Let's use the point \( (2, \frac{7}{5}) \).

Using the slope-intercept form of the equation \( y = mx + b \):

\[ \frac{7}{5} = \frac{2}{5} \cdot 2 + b \]

Calculating the right side:

\[ \frac{7}{5} = \frac{4}{5} + b \]

Now, solving for \( b \):

\[ b = \frac{7}{5} - \frac{4}{5} = \frac{3}{5} \]

### Step 4: Write the equation

Now that we have the slope \( m = \frac{2}{5} \) and the y-intercept \( b = \frac{3}{5} \), we can write the equation that relates \( y \) to \( x \):

\[ y = \frac{2}{5}x + \frac{3}{5} \]

### Step 5: Verify the equation with another point

To ensure that our equation is correct, we can verify it using another point from the table. Let's use \( x = 4 \):

\[ y = \frac{2}{5} \cdot 4 + \frac{3}{5} = \frac{8}{5} + \frac{3}{5} = \frac{11}{5} \]

This matches the value of \( y \) when \( x = 4 \) in the table, confirming that our equation is correct.

### Conclusion

The equation that relates \( y \) to \( x \) according to the values shown in the table is:

\[ y = \frac{2}{5}x + \frac{3}{5} \]