Inverse variation of two variables and the constant of variation
The inverse variation equation
In this lesson we’ll look at solving equations that express inverse variation relationships, which are relationships of the form
???y=\frac{k}{x}???
In an inverse variation relationship you have two variables, usually ???x??? and ???y???, and a constant value that is usually called ???k???.
The main idea in inverse variation is that as one variable increases the other variable decreases. That means that if ???x??? is increasing ???y??? is decreasing, and if ???x??? is decreasing ???y??? is increasing. The number ???k??? is a constant so it’s always the same number throughout the inverse variation problem.
Inverse variation can also be called an “inverse proportion”. The variables ???x??? and ???y??? can be called inversely proportional to one another. The constant of variation is the number ???k???.
An inverse variation can look like, ???xy=k??? or also ???y=k/x???.
The relationship between x and y when they vary inversely, and the constant of variation
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Finding the constant of variation when y varies inversely with x
Example
If you know that ???y??? varies inversely with ???x???, and ???x=10??? when ???y=4???, what is the constant of variation? What is the function for the inverse proportion?
The formula for an inverse variation is ???xy=k???.
We know that ???x=10??? and ???y=4???, so
???10\cdot4=k???
???40=k???
The constant of variation is ???k=40???.
The inverse variation function is therefore
???y=\frac{k}{x}???
???y=\frac{40}{x}???
Let’s look at another example.
Relationship between variables whose product is fixed
Example
The product of two numbers is always ???100???. Describe the relationship between the two numbers as a function.
The equation ???xy=k??? is an example of an inverse variation. It means that when you multiply two numbers together they will always equal the same number.
The question says that ???xy=100???, so the relationship is an inverse relationship.
The function is then
???y=\frac{100}{x}???
because ???100??? is the constant of variation.
Sometimes it can be helpful to think about how to solve two-step equations that are in the form of an inverse variation problem.
Solving a two-step equation when x and y vary inversely
Example
Solve the two-step equation, given that ???x??? and ???y??? vary inversely, ???k/x=y???.
If ???k/5=3??? and ???k/x=15???, find the value of ???x???, when ???y=30???.
We’ll solve the first equation for ???k???.
???\frac{k}{5}=3???
???\frac{k}{5}\cdot5=3\cdot5???
???k=15???
Now we’ll take the value we found for ???k??? and plug it into the second equation to solve for ???x???.
???\frac{15}{x}=30???
???\frac{15}{x}(x)=30(x)???
???15=30x???
???\frac{15}{30}=\frac{30x}{30}???
???\frac{1}{2}=x???