To determine the slope and y-intercept of the linear equation that fits the given points, we can first calculate the slope (\(m\)) using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
We can use any two points from the dataset. Let's use the points (5, 16.5) and (7, 2.5):
- \( (x_1, y_1) = (5, 16.5) \)
- \( (x_2, y_2) = (7, 2.5) \)
Calculating the slope:
\[ m = \frac{2.5 - 16.5}{7 - 5} = \frac{-14}{2} = -7 \]
Next, we can use one of the points and the slope to find the y-intercept (\(b\)) using the equation of a line \(y = mx + b\).
Using the point (5, 16.5):
\[ 16.5 = -7(5) + b \] \[ 16.5 = -35 + b \] \[ b = 16.5 + 35 = 51.5 \]
Hence, the linear equation is:
\[ y = -7x + 51.5 \]
Now, let's double-check using another pair of points, such as (7, 2.5) and (11, 31.5):
Using (7, 2.5) and (11, 31.5):
- \( (x_1, y_1) = (7, 2.5) \)
- \( (x_2, y_2) = (11, 31.5) \)
Calculating the slope:
\[ m = \frac{31.5 - 2.5}{11 - 7} = \frac{29}{4} = 7.25 \]
But since the slope we found previously is already not matching, it seems we have made a calculation error.
To summarize, I will check if the original calculations for multiple points yield a consistent line. If these calculations were performed correctly:
- Various combinations lead to different slopes inconsistent with both given options of y-intercepts.
Thus, the pairs of points do lead me towards checking the outputs correctly.
So, we can summarize:
- The slope and intercept from earlier calculations might not point towards strong fit unless further checked herein the options provided.
None of the given multiple-choice answers match this due to errors in slopes.
If you need further help, please ask!