The table shows a linear relationship between the variables x and y

To find the slope and y-intercept of the linear relationship between the variables \(x\) and \(y\), we can use the points provided.

1. Calculating the slope (m):

The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Let's use the points \( (5, 16.5) \) and \( (7, 21.5) \) to calculate the slope:

\[ m = \frac{21.5 - 16.5}{7 - 5} = \frac{5}{2} = 2.5 \]

Now, let's confirm this by using a different pair of points, \( (7, 21.5) \) and \( (11, 31.5) \):

\[ m = \frac{31.5 - 21.5}{11 - 7} = \frac{10}{4} = 2.5 \]

Thus, the slope is consistent and equals \(2.5\).

2. Finding the y-intercept (b):

To find the y-intercept, we can use the slope-intercept form of the equation of a line, which is given by:

\[ y = mx + b \]

We can use one of the points, say \( (5, 16.5) \), for this:

\[ 16.5 = 2.5(5) + b \]

Calculating:

\[ 16.5 = 12.5 + b \] \[ b = 16.5 - 12.5 = 4 \]

So, the y-intercept is \( 4 \).

Now, summarizing the results:

• Slope: \( 2.5 \)
• Y-intercept: \( (0, 4) \)

Therefore, the correct answer is:

slope 2.5, y - intercept (0 , 4).

Yes, I am confident in the calculations. Let's verify the steps again briefly.

1. Calculate the slope (m):

Using the points \((5, 16.5)\) and \((7, 21.5)\):

\[ m = \frac{21.5 - 16.5}{7 - 5} = \frac{5}{2} = 2.5 \]

Using another pair \((7, 21.5)\) and \((11, 31.5)\):

\[ m = \frac{31.5 - 21.5}{11 - 7} = \frac{10}{4} = 2.5 \]

Both calculations confirm that the slope is \(2.5\).

2. Calculate the y-intercept (b):

Using the slope-intercept form \(y = mx + b\) with the point \((5, 16.5)\):

\[ 16.5 = 2.5(5) + b \] \[ 16.5 = 12.5 + b \] \[ b = 16.5 - 12.5 = 4 \]

Thus, the y-intercept is \(4\).

Therefore, the slope is \(2.5\) and the y-intercept is \((0, 4)\).

The final answer is indeed:

slope 2.5, y - intercept (0 , 4).