Is $x + y = 6$ $x+y=6$ a direct variation and if it is, how do you find the constant?

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Answer 1 · 2017-10-19T11:10:19

$x + y = 6$ $x+y=6$ is not a direct variation

There are several ways to see this:

a direct variation must be convertible into the form $y = c x$ $y=cx$ for some constant $c$ $c$ ; this equation can not be converted in this way.
$(x, y) = (0, 0)$ $(x,y)=(0,0)$ will always be a valid solution for a direct variation; it is not a solution for this equation. [Warning this is a necessary but not sufficient condition i.e. if $(x, y) = (0, 0)$ $(x,y)=(0,0)$ is a solution then the equation might or might not be a direct variation.]
if an equation is a direct variation and $(x, y) = (a, b)$ $(x,y)=(a,b)$ is a solution, then for any constant $c$ $c$ , $(x, y) = (c x, c y)$ $(x,y)=(cx,cy)$ must also be a solution; in this case $(x, y) = (4, 2)$ $(x,y)=(4,2)$ is a solution but $(x, y) = (4 \times 3 = 12, 2 \times 3 = 6)$ $(x,y)=(4xx3=12,2xx3=6)$ is not a solution.

Is x+y=6x+y=6x+y=6 a direct variation and if it is, how do you find the constant?