Difference between revisions of "Modeling using Variation"
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Direct variation describes a simple relationship between two variables . We say y varies directly with x (or as x , in some textbooks) if: | Direct variation describes a simple relationship between two variables . We say y varies directly with x (or as x , in some textbooks) if: | ||
| − | : y = kx | + | :<math> y = kx </math> |
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| − | An inverse variation can be represented by the equation xy = k or <math>y = \frac{k}{x}</math> . | + | An inverse variation can be represented by the equation <math> xy = k </math> or <math>y = \frac{k}{x}</math> . |
| − | That is, y varies inversely as x if there is some nonzero constant k such that, xy = k or <math>y = {k \over x}</math> where x ≠ 0, y ≠ 0 . | + | That is, y varies inversely as x if there is some nonzero constant k such that, <math>xy = k</math> or <math>y = {k \over x}</math> where x ≠ 0, y ≠ 0 . |
==Joint Variation== | ==Joint Variation== | ||
Joint variation occurs when a variable varies directly or inversely with multiple variables. | Joint variation occurs when a variable varies directly or inversely with multiple variables. | ||
| − | For instance, if x varies directly with both y and z, we have x = kyz. If x varies directly with y and inversely with z, we have <math>x = {ky \over z}</math> . | + | For instance, if x varies directly with both y and z, we have <math>x = kyz</math>. If x varies directly with y and inversely with z, we have <math>x = {ky \over z}</math> . |
Notice that we only use one constant in a joint variation equation. | Notice that we only use one constant in a joint variation equation. | ||
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See also: [[Systems of Equations in Two Variables]] | See also: [[Systems of Equations in Two Variables]] | ||
* [https://courses.lumenlearning.com/precalcone/chapter/modeling-using-variation/ Modeling Using Variation], Lumen Learning | * [https://courses.lumenlearning.com/precalcone/chapter/modeling-using-variation/ Modeling Using Variation], Lumen Learning | ||
| + | * [https://opentextbc.ca/intermediatealgebraberg/chapter/2-7-variation-word-problems/ Variation Word Problems], Open Text Intermediate Algebra | ||
* [https://www.khanacademy.org/math/algebra-home/alg-rational-expr-eq-func/alg-direct-and-inverse-variation/v/direct-and-inverse-variation Intro to Direct & Inverse Variation], Khan Academy | * [https://www.khanacademy.org/math/algebra-home/alg-rational-expr-eq-func/alg-direct-and-inverse-variation/v/direct-and-inverse-variation Intro to Direct & Inverse Variation], Khan Academy | ||
* [https://www.youtube.com/watch?v=AMzCEcsd09o Direct Inverse and Joint Variation Word Problems], The Organic Chemistry Tutor | * [https://www.youtube.com/watch?v=AMzCEcsd09o Direct Inverse and Joint Variation Word Problems], The Organic Chemistry Tutor | ||
Revision as of 11:51, 4 October 2021
There are three types of variations:
- Direct variation
- Inverse variation
- Joint variation
Direct Variation
Direct variation describes a simple relationship between two variables . We say y varies directly with x (or as x , in some textbooks) if:
for some constant k , called the constant of variation or constant of proportionality . (Some textbooks describe direct variation by saying " y varies directly as x ", " y varies proportionally as x ", or " y is directly proportional to x .")
This means that as x increases, y increases and as x decreases, y decreases—and that the ratio between them always stays the same.
Inverse Variation
While direct variation describes a linear relationship between two variables , inverse variation describes another kind of relationship.
For two quantities with inverse variation, as one quantity increases, the other quantity decreases.
An inverse variation can be represented by the equation or .
That is, y varies inversely as x if there is some nonzero constant k such that, or where x ≠ 0, y ≠ 0 .
Joint Variation
Joint variation occurs when a variable varies directly or inversely with multiple variables.
For instance, if x varies directly with both y and z, we have . If x varies directly with y and inversely with z, we have .
Notice that we only use one constant in a joint variation equation.
Resources
See also: Systems of Equations in Two Variables
- Modeling Using Variation, Lumen Learning
- Variation Word Problems, Open Text Intermediate Algebra
- Intro to Direct & Inverse Variation, Khan Academy
- Direct Inverse and Joint Variation Word Problems, The Organic Chemistry Tutor
