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### Course: Algebra (all content)>Unit 13

Lesson 7: Direct and inverse variation

# Intro to direct & inverse variation

Sal explains what it means for quantities to vary directly or inversely, and gives many examples of both types of variation. Created by Sal Khan.

## Want to join the conversation?

• This might be a stupid question, but why do we use "k" as the constant?
• I would imnagine it is from the German "konstant."
• Does an inverse variation represent a line? Also, are these directly connected with functions and inverse functions?
• How can π*x be direct variation? Pi is irrational, and keeps going on and on, so there would be no exact scale for both x and y. Thank you for the help!
• The number pi is not going anywhere. It is fixed somewhere between 3 and 4. The y-scale could be indexed by pi itself.
• Why is 4x + 3y = 24 an equation that does not represent direct variation?
• The reason is that y doesn't vary by the same proportion that x does (because of the constant, 24).

In your equation, "y = -4x/3 + 6", for x = 1, 2, and 3, you get y = 4 2/3, 3 1/3, and 2. For x = -1, -2, and -3, y is 7 1/3, 8 2/3, and 10. Notice that as x doubles and triples, y does not do the same, because of the constant 6.

To quote zblakley from his answer here 5 years ago:
"The difference between the values of x and y is not what dictates whether the variation is direct or inverse. What is important is the factor by which they vary. While y becomes more negative as x becomes more positive, they will still vary by the same factor (i.e. if you increase x from 1 to 4 that's a factor of 4, the value of y [in y = -2x] will go from -2 (when x=1) to -8 (when x=4) which is also a factor of 4).
• I see comments about problems in a practice section. Besides the 3 questions about recognizing direct and inverse variations, are there practice problems anywhere?
• At about , (when talking about direct variation) Sal says that "in general... if y varies directly with x... x varies directly with y." Are there any cases where this is not true? In other words, are there any cases when x does not vary directly with y, even when y varies directly with x?
Thanks!
• I don't get what varies means? Can someone tell me.
• "Varies" refers to how one variable changes in relationship to the other variable.
• i know this is a wierd question but what do you do when in a direct variation when your trying to find K what do you do when X wont go into Y evenly? do you just use decimal form or fraction form?
• You can use the form that you prefer; the two are equivalent. In the Khan A. exercises, accepted answers are simplified fractions and decimal answers (except in some exercises specifically about fractions and decimals). If you're not sure of the format to use, click on the "Accepted formats" button at the top right corner of the answer box.
• At , where you give the formula for inverse variation, I am confused. The formula that my teacher gave us was ( y = k/x ) Please help and thanks so much!!
• Both your teacher's equation ( y = k / x) and Sal's equation ( y = k * 1/x) mean the same thing, like they will equal the same number. To show this, let's plug in some numbers.

How about x = 2 and k = 4?

``y = k / xy = 4 / 2y = 2``

and now Sal's:

``y = k * 1/xy = 4 * 1/2 (Since we know 1/2 equals .5, let's use that instead, usually people understand decimals better for multiplying, but it means the exact same as 1/2)y = 4 * .5(We are essentially taking half of 4)y = 2``

Would you like me to explain why? It takes a bit of explaining on fractions and how they work :)

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If you want to see how we would multiply 4 * 1/2, here's a picture I drew to explain it =

http://i.imgur.com/EkViZOw.png