1. A line, y = mx + b, passes through the point (1,6) and is parallel to y = 4x + 6. What is the value of b?
2. A set of nine data points is shown below. Which statement is true if a tenth data point of 45 is added to the data set?
8 11 12 10 9 7 5 3 9
A. The mean and median will both increase.
B. The mean will increase and the median will decrease.
C. The mean will increase and the median will remain the same.
D. The mean and the median will both decrease.
y = mx+b, so plug in your point:
6 = 4*1 + b
...
since 45 is larger than any of the existing values, the mean will rise.
since there are no duplicate values, the median will also rise.
(note that if the two center values were equal, the median might stay the same - why?
Thank you! The median might stay the same because they are the same?
1. To find the value of b in the equation y = mx + b, we need to use the fact that the line is parallel to y = 4x + 6.
Since the given line is parallel to y = 4x + 6, it means they have the same slope. The slope of y = 4x + 6 is 4, which means the slope of the parallel line is also 4.
Now that we know the slope of the line is 4, we can use the point (1,6) to find the value of b.
Substitute the coordinates (1,6) into the equation y = 4x + b:
6 = 4(1) + b
6 = 4 + b
Subtract 4 from both sides:
6 - 4 = b
2 = b
So, the value of b is 2.
2. To determine the effect of adding a data point of 45 to the existing dataset, we need to consider the impact on the mean and median.
The mean of a dataset is found by summing up all the numbers and dividing by the total count. The median is the middle value when the dataset is arranged in ascending order.
In this case, let's calculate the mean and median before and after adding the number 45 to the dataset:
Before adding 45 (original dataset): 8, 11, 12, 10, 9, 7, 5, 3, 9
Mean (original): (8 + 11 + 12 + 10 + 9 + 7 + 5 + 3 + 9) / 9 = 74 / 9 ≈ 8.22
Median (original): Arrange the numbers in ascending order: 3, 5, 7, 8, 9, 9, 10, 11, 12. The median is 9.
After adding 45 (modified dataset): 8, 11, 12, 10, 9, 7, 5, 3, 9, 45
Mean (modified): (8 + 11 + 12 + 10 + 9 + 7 + 5 + 3 + 9 + 45) / 10 = 119 / 10 = 11.9
Median (modified): Arrange the numbers in ascending order: 3, 5, 7, 8, 9, 9, 10, 11, 12, 45. The median is 9.
Based on the calculations, we can see that the mean increases from approximately 8.22 to 11.9, so the mean will increase by adding the number 45.
However, the median remains the same at 9, so the median will remain the same after adding the number 45.
Therefore, the correct answer is C. The mean will increase and the median will remain the same.