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Question:
What is the equation of the line of best fit?
Solution:
By definition, the equation of the line is given by:
y = mx+b
where m is the slope of the line and b is the y-coordinate of the y-intercept. Now, by definition the slope of the line is given by the following equation:
[tex]m\text{ = }\frac{Y2-Y1}{X2-X1}[/tex]where (X1,Y1) and (X2,Y2) are points on the line (in our case points on the table). Taking into account this, take any two points on the table, for example
(X1,Y1) = (-2,2)
(X2,Y2) = (1,1)
Put these points in the equation for the slope:
[tex]m\text{ = }\frac{Y2-Y1}{X2-X1}=\text{ }\frac{1-2}{1-(-2)}=\text{ }\frac{-1}{1+2}=\frac{-1}{3}[/tex]then, the slope of the line is -1/3. Put this value on the equation of the line we get:
[tex]y\text{ = }\frac{-1}{3}x\text{ + b}[/tex]now, we want to find b. For this, take any two points in the table, and put these points in the previous equation. Take for example (x,y) =(0,1), then we have:
[tex]1\text{= }\frac{-1}{3}(0)\text{ + b}[/tex]then, we can conclude that b = 1. Thus, we can conclude that the equation of the line is:
[tex]y\text{ = }\frac{-1}{3}x\text{ +}1[/tex]
