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.
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:
The following are equivalent:
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:
The following are equivalent:
By combining phrases, a wide variety of relationships can be described between variables.
For example:
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. |
PROBLEM TYPES:
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