Step by step
The question
Question: y varies directly with the square of x.
y = 20 when x = 2.
Find y when x = 5.
1 Write the rule
Varies directly means y is always the same number times x².
When x gets bigger, y gets bigger too.
We call the number k: y = kx².
k stays the same for every pair of numbers.
Books call it the constant of variation.
2 Find k
We put in y = 20 and x = 2.
2 squared is 2 times 2: 4.
And k times 4 is written 4k.
4k means 4 times k.
We want just k.
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
Careful!
k is y divided by x²: 20/4.
Not 4/20.
The y number goes on top.
Is k alone now?
Not yet: 4k means 4 times k.
How do we remove it?
20/4 on the left.
4k/4 on the right.
4 divided by 4 is 1.
They cancel each other.
So just k is left.
20 divided by 4 is 5.
So k = 5.
So after step 2: we divided by 4, and now we have k = 5.
3 Write the equation
We put 5 in place of k.
So the equation is y = 5x².
4 Find y
We put in x = 5.
5 squared is 25.
5 times 25 is 125.
So y = 125.
5 Check
Let's check.
In direct variation, y divided by x² is always k.
That is how we found k.
First pair: 20/4 is 5.
New pair: 125/25 is 5 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.