Step by step
The question
Question: y varies inversely with the square of x.
y = 4 when x = 3.
Find y when x = 2.
1 Write the rule
Varies inversely means y is a number divided by x².
When x gets bigger, y gets smaller.
We call the number k: y = k/x².
k stays the same for every pair of numbers.
Books call it the constant of variation.
2 Find k
We put in y = 4 and x = 3.
3 squared is 9.
k/9 means k divided by 9.
We want just k.
To remove divided by 9, we multiply by 9.
We do it on both sides, so they stay equal.
Divided by 9, then times 9: they cancel.
Cross them out.
4 times 9 is 36.
So k = 36.
3 Write the equation
We put 36 in place of k.
So the equation is y = 36/x².
4 Find y
We put in x = 2.
2 squared is 4.
Careful!
Inverse is not direct.
x got smaller, from 3 to 2.
So y must get bigger.
36 divided by 4 is 9.
So y = 9.
5 Check
Let's check.
In inverse variation, x² times y is always k.
First pair: 3² times 4 is 36.
New pair: 2² times 9 is 36 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.