I have 4 problems like this one can someone complete this one so I can get a reference for the rest? Thank you.

Question

Find the best fitting line to the given data points.

(1,2), (2,3), (3,5)

Answer 1

step 1: Find the slope of the line of best fit (this is the hardest step and after you finish this its smooth sailing)

1. start by calculating the average of the x's and y's
2. let x_a denote the average of the x's: X_a= the sum of all the x's/ the number of x's = (1+2+3)/3= 2
3. let Y_a denote the average of the y's: Y_a= the sum of all the Y's/ the number of Y's = (2+3+5)/3= 10/3
4. Then, the formula for the slope of the line of best fit is the sum of each (x-x_a)(y-y_a)/the sum of each (x-x_a)(x-x_a)
5. Calculate each (x-x_a) and (y-y_a)
6. (1,2): (x₁-x_a)= 1- 2= -1 and (y₁-y_a) = (2- 10/3) = -4/3
7. (2,3): (x₂-x_a)= 2-2 =0 and (y₂-y_a) =(3 -10/3)= -1/3
8. (3,5): (x₃-x_a)= 3-2=1 and (y₃-y_a) =(5- 10/3)= 5/3
9. Calculate each (x-x_a)(x-x_a) and (x-x_a)(y-y_a)
10. (1,2): (x₁-x_a)(x₁-x_a)= (-1)(-1)=1 and (x₁-x_a)(y₁-y_a) = (-1)(-4/3)=4/3
11. (2,3): (x₂-x_a)(x₂-x_a)= (0)(0)= 0 and (x₂-x_a)(y₂-y_a) = (0)(-1/3)=0
12. (3,5): (x₃-x_a)(x₃-x_a)= (1)(1)=1 and (x₃-x_a)(y₃-y_a) = (1) (5/3)=5/3
13. Calculate the sum of every (x-x_a)(x-x_a)
14. [(x₁-x_a)(x₁-x_a)] + [(x₂-x_a)(x₂-x_a)] + [(x₃-x_a)(x₃-x_a)] = 1+0+1=2
15. Calculate the sum of every (x-x_a)(y-y_a)
16. [(x₁-x_a)(y₁-y_a)]+[(x₂-x_a)(y₂-y_a)]+[(x₃-x_a)(y₃-y_a)] = 4/3+0+5/3 = 9/3=3
17. Calculate the slope: [(x₁-x_a)(y₁-y_a)]+[(x₂-x_a)(y₂-y_a)]+[(x₃-x_a)(y₃-y_a)] / [(x₁-x_a)(x₁-x_a)] + [(x₂-x_a)(x₂-x_a)] + [(x₃-x_a)(x₃-x_a)] = 3/2

step 2: calculate the y-intercept using the formula y_a= a(x_a)-b where b= to the y-intercept and a=slope

1. y_a= a(x_a)-b: 10/3= a(2)-b
2. plug in slope
3. 10/3= (3/2)(2)- b
4. solve for b
5. b= - (1/3)

step 3: plug the slope and y-intercept into y= a(x)-b

1. y=(3/2)x - (-1/3) = (3/2)x+1/3