Step by step
The question
Question: y varies directly with the square of x.
y = 12 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 = 12 and x = 2.
2 squared is 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: 12/4.
Not 4/12.
The y number goes on top.
12/4 on the left.
4k/4 on the right.
4 divided by 4 is 1.
So just k is left.
12 divided by 4 is 3.
So k = 3.
3 Write the equation
We put 3 in place of k.
So the equation is y = 3x².
4 Find y
We put in x = 5.
5 squared is 25.
3 times 25 is 75.
So y = 75.
5 Check
Let's check.
In direct variation, y divided by x² is always k.
First pair: 12/4 is 3.
New pair: 75/25 is 3 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.