Step by step
The question
Question: y varies inversely with x.
y = 9 when x = 2.
Find y when x = 6.
1 Write the rule
Varies inversely means y is a number divided by x.
When x gets bigger, y gets smaller.
We call the number k: y = k over x.
k stays the same for every pair of numbers.
Books call it the constant of variation.
2 Find k
We put in y = 9 and x = 2.
k/2 means k divided by 2.
We want just k.
To remove divided by 2, we multiply by 2.
We do it on both sides, so they stay equal.
Divided by 2, then times 2: they cancel.
Cross them out.
9 times 2 is 18.
So k = 18.
3 Write the equation
We put 18 in place of k.
So the equation is y = 18 over x.
4 Find y
We put in x = 6.
Careful!
Inverse is not direct.
x got bigger, from 2 to 6.
So y must get smaller.
18 divided by 6 is 3.
So y = 3.
5 Check
Let's check.
In inverse variation, x times y is always k.
First pair: 2 times 9 is 18.
New pair: 6 times 3 is 18 too.
The same k.
So the answer is right!
Let's look back.
First: write the rule.
Then: find k.
Then: write the equation.
Then: find y.
Last: check.