DIRECT AND INVERSE VARIATION

LESSON READ-THROUGH
by Dr. Carol JVF Burns (website creator)
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The earlier lesson Getting Bigger? Getting Smaller? introduces the concepts of direct and inverse variation.
Study it first, being sure to click-click-click several exercises at the bottom to check your understanding.
This current lesson builds on these prior concepts.

DIRECT VARIATION equivalent statements

The following are equivalent:

  • $y = kx\,$, for $\,k\ne 0$
  • $y\,$ varies directly as $\,x$
  • $y\,$ is directly proportional to $\,x$
  • $y\,$ is proportional to $\,x$
The nonzero constant $\,k\,$ is called the constant of proportionality.

When you are told that ‘$y\,$ is proportional to $x\,$’ then direct variation is being described.
To emphasize this fact, the extra word ‘directly’ can be inserted:   ‘$y\,$ is directly proportional to $\,x$’.

Recall that variables that are directly proportional ‘follow each other in size’:
when one gets bigger (farther away from zero), so does the other;
when one gets smaller (closer to zero), so does the other.

Notice:   if $\,y\,$ is proportional to $\,x\,$, then $\,x\,$ is proportional to $\,y$.
Thus, we can simply say   ‘$x\,$ and $\,y$ are proportional’ or ‘$y\,$ and $\,x$ are proportional’.
Why is this? Study the following list of equivalent sentences:

$y\,$ is proportional to $\,x\,$ given; assumed to be true
$y = kx\,$, for $\,k\ne 0$ an equivalent statement of direct variation; see the list above
$x = \frac 1k y\,$, for $\,\frac 1k\ne 0$ multiplication property of equality (divide both sides by $\,k\ne 0\,$);
also, $\,k\,$ is nonzero if and only if its reciprocal is nonzero
$x\,$ is proportional to $\,y\,$ an equivalent statement of direct variation; see the list above

Also notice:

Next, we talk about inverse variation:

INVERSE VARIATION equivalent statements

The following are equivalent:

  • $\displaystyle y = \frac{k}{x}\,$, for $\,k\ne 0$
  • $y\,$ varies inversely as $\,x$
  • $y\,$ is inversely proportional to $\,x$

Recall that variables that are inversely proportional have sizes that ‘go in opposite directions’:
when one gets bigger, the other gets smaller;
when one gets smaller, the other gets bigger.

Similar to the argument above:   if $\,y\,$ is inversely proportional to $\,x\,$, then $\,x\,$ is inversely proportional to $\,y$.
Thus, we can simply say   ‘$x\,$ and $\,y$ are inversely proportional’ or ‘$y\,$ and $\,x$ are inversely proportional’.

Also notice:

Sometimes it is necessary to talk about a relationship between more than two variables:

JOINT PROPORTIONALITY equivalent statements

The following are equivalent:

  • $\displaystyle z = kxy\,$, for $\,k\ne 0$
  • $z\,$ varies jointly as $\,x\,$ and $\,y$
  • $z\,$ is jointly proportional to $\,x\,$ and $\,y\,$

By combining phrases, a wide variety of relationships can be described between variables.
For example:

Master the ideas from this section
by practicing the exercise at the bottom of this page.


When you're done practicing, move on to:
Proportionality Problems

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2
AVAILABLE MASTERED IN PROGRESS