For the following exercises, write an equation describing...

For the following exercises, write an equation describing the relationship of the given variables.
$y$ varies jointly as $x$ and $z$ and inversely as $w .$ When $x=3$ , $z=5,$ and $w=6,$ then $y=10 .$

Key Concepts

Joint Variation

Joint variation describes a relationship where one variable depends directly on the product of two or more other variables. When a variable varies jointly as several variables, it means that the value of this variable increases or decreases proportionally to the product of those variables.

Inverse Variation

Inverse variation defines a relationship in which one variable is proportional to the reciprocal of another variable. This indicates that as one variable increases, the other decreases proportionally, and vice versa.

Constant of Proportionality

The constant of proportionality is a fixed number that relates the variables in a variation equation. It is determined by substituting known values into the equation, allowing the formulation of a precise relationship between the variables.

Transcript

00:01 All right.

00:02 So now we're getting into the really good stuff because these problems aren't just varying in one way anymore.

00:07 They're varying with joint and inverse variation.

00:11 So it's like the mother of all problems.

00:13 So when i'm told that y is varying jointly as some variables and inversely as w, i have to take all the knowledge i have of inverse and joint variation and put it all together.

00:23 So my generic version is always y equals since it's y varies.

00:29 And then it's varying joint.

00:31 With x and z so that's a product situation with k my constant k is always the constant i'm going to be finding so here's my joint setup these are varying jointly and then i also have inversely as w so traditionally with inversely i would have k over w i already have the k so all i have to do is divide by w so if it's varying jointly in this way i'm going to take the values they gave me for x is 3, z which is 5, w which is 6, and y which is 10, and i'm going to sub all these into the generic equation in order to find k as a specific value to represent the relationship between these variables.

01:23 And once i have k, i'm going to rewrite the equation so that it specifically represents the relationship between all these variables that are varying both jointly and inversely.

01:34 So here we go.

01:36 I'm going to go, y, which is 10, equals k, which is my unknown, x, which is 3, z which is 5, and w, which is 6.

01:52 I'm going to go ahead and simplify it down...

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