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Problem: Fit cubic equation {{1996,1997,1998,1999,2000},{40,50,62,58,60}} [ Calculator, Method and examples ]

Solution:
Your problem -> Fit cubic equation {{1996,1997,1998,1999,2000},{40,50,62,58,60}}

The equation is y = a + bx + cx^2 + dx^3 and the normal equations are

sum y = an + b sum x + c sum x^2 + d sum x^3

sum xy = a sum x + b sum x^2 + c sum x^3 + d sum x^4

sum x^2y = a sum x^2 + b sum x^3 + c sum x^4 + d sum x^5

sum x^3y = a sum x^3 + b sum x^4 + c sum x^5 + d sum x^6

 X y x = X - 1998 x^2 x^3 x^4 x^5 x^6 x*y x^2*y x^3*y 1996 40 -2 4 -8 16 -32 64 -80 160 -320 1997 50 -1 1 -1 1 -1 1 -50 50 -50 1998 62 0 0 0 0 0 0 0 0 0 1999 58 1 1 1 1 1 1 58 58 58 2000 60 2 4 8 16 32 64 120 240 480 --- --- --- --- --- --- --- --- --- --- --- 9990 270 0 10 0 34 0 130 48 508 168

Substituting these values in the normal equations
270=5a+0b+10c+0d

48=0a+10b+0c+34d

508=10a+0b+34c+0d

168=0a+34b+0c+130d

Solving these 4 equations,
Total Equations are 4

5a+0b+10c+0d=270 -> (1)

0a+10b+0c+34d=48 -> (2)

10a+0b+34c+0d=508 -> (3)

0a+34b+0c+130d=168 -> (4)

Select the equations (1) and (3), and eliminate the variable a.

 5a+10c=270  xx 2->  10a + 20c = 540  − 10a+34c=508  xx 1->  10a + 34c = 508  - 14c = 32  -> (5)

Select the equations (2) and (4), and eliminate the variable b.

 10b+34d=48  xx 17->  170b + 578d = 816  − 34b+130d=168  xx 5->  170b + 650d = 840  - 72d = -24  -> (6)

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