Step by step
The question
Question: y varies directly with the square of x.
y = 50 when x = 5.
Find y when x = 3.
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 = 50 and x = 5.
5 squared is 25, and k times 25 is written 25k.
25k means 25 times k.
We want just k.
To remove times 25, we divide by 25.
We do it on both sides, so they stay equal.
Careful!
k is y divided by x²: 50/25.
Not 25/50.
The y number goes on top.
50/25 on the left.
25k/25 on the right.
25 divided by 25 is 1.
So just k is left.
50 divided by 25 is 2.
So 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 = 3.
3 squared is 9.
2 times 9 is 18.
So y = 18.
5 Check
Let's check.
In direct variation, y divided by x² is always k.
First pair: 50/25 is 2.
New pair: 18/9 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.