The best-fitting line using the least squares method for the given points is y = 2x + 1
To find the best-fitting line using the least squares method for these points, use linear regression.
The equation for a line is y = mx + b
where; (m) is the slope
(b) is the y-intercept.
Calculate the slope (m) and y-intercept (b):
m = n(∑xy) - (∑x)(∑y) / n(∑x²) - (∑x)²
b = (∑y) - m(∑x) / n
Where;
(n) is the number of data points
(∑xy) is the sum of the product of each x and y
(∑x) is the sum of x values
(∑y) is the sum of y values
(∑x²) is the sum of x squared.
Calculating the values:
∑x = 1 + 2 + 3 + 4 = 10
∑y = 3 + 5 + 7 + 9 = 24
∑xy = (1 x 3) + (2 x 5) + (3 x 7) + (4 x 9) = 3 + 10 + 21 + 36 = 70
∑x² = (1²) + (2²) + (3²) + (4²) = 1 + 4 + 9 + 16 = 30
Using these values:
m = (4 x 70) - (10 x 24) / (4 x 30) - 10²
m = 280 - 240 / 120 - 100
m = 40 / 20 = 2
b = 24 - m x 10 / 4
b = 24 - (2 x 10) / 4
m = 4 / 4} = 1
Therefore, the best-fitting line using the least squares method for these points is y = 2x + 1