How do you determine the constant of variation for the direct variation given by (1,8) (2,4) (4,2) (8,1)?

1 Answer
Alan P.
Oct 10, 2015

The set given does not represent a direct variation;
it is an inverse variation with a constant of 8

Explanation:

If (x,y) in direct variation set
rarr y=c*x for some constant c and all pairs (x,y)
If we consider only the first 2 pairs of the defining set
we have:
color(white)("XXX")8=c*1
and
color(white)("XXX")4=c*2
Obviously both can not be true for any single value of c.

However, if y varies inversely with x then
rarr x*y = c for some constant c and all pairs (x,y)
and looking at all given (x,y) pairs:
(1,8)rarr 1xx8 = 8
(2,4)rarr 2xx4 = 8
(4,2)rarr 4xx2 = 8
(8,1)rarr 8xx1 = 8
We appear to have an inverse variation with a constant of 8

For a direct variation if you multiply the value of one of your variables by a number, it must result in the value of the other variable being multiplied by that same number.

For an inverse variation if you multiply the value of one of your variables by a number, it must result in the value of the other variable being divided by that number.

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