Question
Find the equation of the line of best fit for the points (-4, 10), (-1, 5), (2, -1), (3, -6),
and (5, -7).
answers:
y =- 2x+2
y=2x -2
y=2r - 0.98
y =- 2x- 0.98
and (5, -7).
answers:
y =- 2x+2
y=2x -2
y=2r - 0.98
y =- 2x- 0.98
Answers
Answered by
GPT-4o mini
To find the equation of the line of best fit for the given points, we can use the least squares method. The formula for the slope (\(m\)) of the line of best fit is given by:
\[
m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2}
\]
And the formula for the y-intercept (\(b\)) is:
\[
b = \frac{\sum y - m(\sum x)}{N}
\]
where \(N\) is the number of points, \(\sum xy\) is the sum of the product of each x and y coordinate, \(\sum x\) is the sum of x coordinates, \(\sum y\) is the sum of y coordinates, and \(\sum x^2\) is the sum of the squares of x coordinates.
Let's calculate the necessary values for these points \((-4, 10), (-1, 5), (2, -1), (3, -6), (5, -7)\).
1. Calculate \(N\):
\[
N = 5
\]
2. Calculate \(\sum x\), \(\sum y\), \(\sum xy\), and \(\sum x^2\):
\[
\sum x = -4 + (-1) + 2 + 3 + 5 = 5
\]
\[
\sum y = 10 + 5 + (-1) + (-6) + (-7) = 1
\]
\[
\sum xy = (-4)(10) + (-1)(5) + (2)(-1) + (3)(-6) + (5)(-7)
\]
Calculating each term:
\(-4 \times 10 = -40\)
\(-1 \times 5 = -5\)
\(2 \times -1 = -2\)
\(3 \times -6 = -18\)
\(5 \times -7 = -35\)
Now summing these products:
\[
\sum xy = -40 - 5 - 2 - 18 - 35 = -100
\]
\[
\sum x^2 = (-4)^2 + (-1)^2 + (2)^2 + (3)^2 + (5)^2
\]
Calculating each term:
\((-4)^2 = 16\)
\((-1)^2 = 1\)
\((2)^2 = 4\)
\((3)^2 = 9\)
\((5)^2 = 25\)
Now summing these:
\[
\sum x^2 = 16 + 1 + 4 + 9 + 25 = 55
\]
3. Now substitute into the slope formula:
\[
m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2}
\]
Substituting in the values:
\[
m = \frac{5(-100) - (5)(1)}{5(55) - (5)^2}
\]
Simplifying:
\[
m = \frac{-500 - 5}{275 - 25} = \frac{-505}{250} = -2.02
\]
4. Now find the y-intercept \(b\):
\[
b = \frac{\sum y - m(\sum x)}{N}
\]
Substituting in the values:
\[
b = \frac{1 - (-2.02)(5)}{5}
\]
Calculating:
\[
b = \frac{1 + 10.1}{5} = \frac{11.1}{5} = 2.22
\]
5. Thus, the line of best fit is:
\[
y = -2.02x + 2.22
\]
Since none of the answers provided directly match this result, we take a closer look. The slope approximately \(m \approx -2\) and \(b \approx 2\), indicating that the closest option would be:
\[
y = -2x + 2
\]
Hence, the answer is:
\[
\boxed{y = -2x + 2}
\]
\[
m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2}
\]
And the formula for the y-intercept (\(b\)) is:
\[
b = \frac{\sum y - m(\sum x)}{N}
\]
where \(N\) is the number of points, \(\sum xy\) is the sum of the product of each x and y coordinate, \(\sum x\) is the sum of x coordinates, \(\sum y\) is the sum of y coordinates, and \(\sum x^2\) is the sum of the squares of x coordinates.
Let's calculate the necessary values for these points \((-4, 10), (-1, 5), (2, -1), (3, -6), (5, -7)\).
1. Calculate \(N\):
\[
N = 5
\]
2. Calculate \(\sum x\), \(\sum y\), \(\sum xy\), and \(\sum x^2\):
\[
\sum x = -4 + (-1) + 2 + 3 + 5 = 5
\]
\[
\sum y = 10 + 5 + (-1) + (-6) + (-7) = 1
\]
\[
\sum xy = (-4)(10) + (-1)(5) + (2)(-1) + (3)(-6) + (5)(-7)
\]
Calculating each term:
\(-4 \times 10 = -40\)
\(-1 \times 5 = -5\)
\(2 \times -1 = -2\)
\(3 \times -6 = -18\)
\(5 \times -7 = -35\)
Now summing these products:
\[
\sum xy = -40 - 5 - 2 - 18 - 35 = -100
\]
\[
\sum x^2 = (-4)^2 + (-1)^2 + (2)^2 + (3)^2 + (5)^2
\]
Calculating each term:
\((-4)^2 = 16\)
\((-1)^2 = 1\)
\((2)^2 = 4\)
\((3)^2 = 9\)
\((5)^2 = 25\)
Now summing these:
\[
\sum x^2 = 16 + 1 + 4 + 9 + 25 = 55
\]
3. Now substitute into the slope formula:
\[
m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2}
\]
Substituting in the values:
\[
m = \frac{5(-100) - (5)(1)}{5(55) - (5)^2}
\]
Simplifying:
\[
m = \frac{-500 - 5}{275 - 25} = \frac{-505}{250} = -2.02
\]
4. Now find the y-intercept \(b\):
\[
b = \frac{\sum y - m(\sum x)}{N}
\]
Substituting in the values:
\[
b = \frac{1 - (-2.02)(5)}{5}
\]
Calculating:
\[
b = \frac{1 + 10.1}{5} = \frac{11.1}{5} = 2.22
\]
5. Thus, the line of best fit is:
\[
y = -2.02x + 2.22
\]
Since none of the answers provided directly match this result, we take a closer look. The slope approximately \(m \approx -2\) and \(b \approx 2\), indicating that the closest option would be:
\[
y = -2x + 2
\]
Hence, the answer is:
\[
\boxed{y = -2x + 2}
\]