Step by step
The question
Question: y varies directly with the square of x.
y = 18 when x = 3.
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 = 18 and x = 3.
3 squared is 3 times 3: 9.
And k times 9 is written 9k.
9k means 9 times k.
We want just k.
To remove times 9, we divide by 9.
We do it on both sides, so they stay equal.
Careful!
k is y divided by x²: 18/9.
Not 9/18.
The y number goes on top.
Is k alone now?
Not yet: 9k means 9 times k.
How do we remove it?
18/9 on the left.
9k/9 on the right.
9 divided by 9 is 1.
They cancel each other.
So just k is left.
18 divided by 9 is 2.
So k = 2.
So after step 2: we divided by 9, and now we have k = 2.
3 Write the equation
We put 2 in place of k.
So the equation is y = 2x².
4 Find y
We put in x = 5.
5 squared is 25.
2 times 25 is 50.
So y = 50.
5 Check
Let's check.
In direct variation, y divided by x² is always k.
That is how we found k.
First pair: 18/9 is 2.
New pair: 50/25 is 2 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.