Can Negative Direct Variation be Considered Direct Variation?

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SUMMARY

Direct variation is defined by the equation y = kx, where k represents the constant of proportionality. This discussion clarifies that k can indeed be negative, resulting in a negative slope, which still maintains direct variation. As x increases, y decreases when k is negative, yet the ratio of y to x remains constant, confirming direct proportionality. The assertion that direct variation requires k to be positive is incorrect.

PREREQUISITES
  • Understanding of direct variation and its mathematical representation (y = kx).
  • Knowledge of the concept of proportionality in mathematics.
  • Familiarity with the implications of positive and negative slopes in linear equations.
  • Basic algebra skills for manipulating equations and understanding ratios.
NEXT STEPS
  • Study the implications of negative constants in linear equations.
  • Explore the concept of direct variation in different mathematical contexts.
  • Learn about the graphical representation of direct variation and how slopes affect it.
  • Investigate common misconceptions in mathematical definitions and their corrections.
USEFUL FOR

Students studying algebra, educators teaching mathematical concepts, and anyone seeking to clarify the principles of direct variation and proportionality.

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Homework Statement



Alright, so I just want some clarification on direct variation, since it seems that every internet source I can find is (seemingly to me) wrong.

To me, direct variation means that the ratio of y to x is fixed with y=kx where k is the constant of proportionality.

Various sources online say that this means as y increases so does x or as y decreases so does x. BUT, I think this is wrong for the following reason.

If k is negative such that y=-kx, then as x increases y will DECREASE. However, if you take a bunch of co-ordinates and find the proportion of y to x, you get a negative constant (-k). Therefore they must be directly proportional. I think of this as a negative slope to a line, the line shows that the co-ordinates are proportional, but as x increases, y decreases.

So am I correct? Can the constant of proportionality k be negative and show direct variation? If not why not?

Homework Equations



y=kx, y=-kx

The Attempt at a Solution



(see above)
 
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"Can the constant of proportionality k be negative and show direct variation?"

Yes. Y is directly proportional to x if there is a constant k such that Y = kX. NO mention need be made of whether k is positive or negative.
 
Alright, thanks for the help! There seems to be an awful lot of technically incorrect online resources on this. :P
 

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