Calculate Fitting straight line - Curve fitting using Least square method

X	Y
3	2.3
5	2.6
7	2.8
9	3.2
11	3.5

Solution:
Method-1 of solution :
Straight line equation is `y = a + bx`.

The normal equations are
`sum y = an + b sum x`

`sum xy = a sum x + b sum x^2`


The values are calculated using the following table

`x`	`y`	`x^2`	`x*y`
3	2.3	9	6.9
5	2.6	25	13
7	2.8	49	19.6
9	3.2	81	28.8
11	3.5	121	38.5
---	---	---	---
`sum x=35`	`sum y=14.4`	`sum x^2=285`	`sum x*y=106.8`

Substituting these values in the normal equations
`5a+35b=14.4`

`35a+285b=106.8`


Solving these two equations using Elimination method,
`5a+35b=14.4`

and `35a+285b=106.8`

`5a+35b=14.4 ->(1)`

`35a+285b=106.8 ->(2)`

equation`(1) xx 7 =>35a+245b=100.8`

equation`(2) xx 1 =>35a+285b=106.8`

Substracting `=>-40b=-6`

`=>40b=6`

`=>b=6/40`

`=>b=3/20`

`=>b=0.15`

Putting `b=0.15` in equation `(1)`, we have

`5a+35(0.15)=14.4`

`=>5a=14.4-5.25`

`=>5a=9.15`

`=>a=9.15/5`

`=>a=1.83`

`:.a=1.83" and "b=0.15`

Now substituting this values in the equation is `y = a + bx`, we get

`y = 1.83 +0.15x`

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Method-2 of solution :

Equation of straight line is `y=mx+b`, where Slope is m and Intercept is b

`m=(n sum xy - sum x sum y) / (n sum(x^2) - (sum x)^2)`

`b=(sum y - m sum x)/n`

The values are calculated using the following table

`x`	`y`	`x^2`	`x*y`
3	2.3	9	6.9
5	2.6	25	13
7	2.8	49	19.6
9	3.2	81	28.8
11	3.5	121	38.5
---	---	---	---
`sum x=35`	`sum y=14.4`	`sum x^2=285`	`sum x*y=106.8`

Find the value of Slope `m`

`m=(n sum xy - sum x sum y) / (n sum(x^2) - (sum x)^2)`

`:.m=(5 * 106.8 - 35*14.4) / (5* 285 - (35)^2)`

`:.m=(534 - 504) / (1425 - 1225)`

`:.m=(30) / (200)`

`:.m=0.15`

Find the value of Intercept `b`

`b=(sum y - m sum x)/n`

`:.b=(14.4 - 0.15 * 35)/5`

`:.b=(14.4 -5.25)/5`

`:.b=(9.15)/5`

`:.b=1.83`

So the required equation is `y=mx+b`

`y=0.15x+1.83`

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The (x,y) points and line `y = 1.83 +0.15x` on a graph

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