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

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

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Answer 1 · 2015-10-10T10:26:17

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

Explanation:

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

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

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

1 Answer

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