Question 1.5.2: Point Constraint Consider the data given at the beginning of...

System Dynamics

Point Constraint

Consider the data given at the beginning of this section

x	0	5	10
y	2	6	11

We found that the best-fit line is y = (9/10)x + 11/6. Find the best-fit line that passes through the point x = 10, y = 11.

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Subtracting 10 from all the x values and 11 from all the y values, we obtain a new set of data in terms of the new variables X = x − 10 and Y = y − 11.

X	-10	-5	0
Y	-9	-5	0

Expressing (1.5.3) in terms of

$m \sum^{n}_{i=1} {X^{2}_{i}} = \sum^{n}_{i=1} {X_{i} Y_{i}}$ (1.5.3)

m  \sum^{3}_{i=1}  {X^{2}_{i}} = \sum^{3}_{i=1}   {X_{i} Y_{i}}

\sum^{3}_{i=1}  {X^{2}_{i} = (−10)^{2} + 5^{2} + 0 = 125 }

$\sum^{3}_{i=1} {X_{i} Y_{i} = (−10)(−9) + (−5)(−5) + 0 = 115 }$
Thus, m = 115/125 = 23/25 and the best-fit line is Y = (23/25)X. In terms of the original variables, this line is expressed as y − 11 = (23/25)(x − 10) or y = (23/25)x + 9/5.

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