Step by step
The question
Question: y varies directly with x.
y = 15 when x = 3.
Find y when x = 8.
1 Write the rule
Varies directly means y is always the same number times x.
When x gets bigger, y gets bigger too.
We call the number k: y = kx.
k stays the same for every pair of numbers.
Books call it the constant of variation.
2 Find k
We put in y = 15 and x = 3.
k times 3 is written 3k.
3k means 3 times k.
We want just k.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
Careful!
k is y divided by x: 15/3.
Not 3/15.
The y number goes on top.
15/3 on the left.
3k/3 on the right.
3 divided by 3 is 1.
So just k is left.
15 divided by 3 is 5.
So k = 5.
3 Write the equation
We put 5 in place of k.
So the equation is y = 5x.
4 Find y
We put in x = 8.
5 times 8 is 40.
So y = 40.
5 Check
Let's check.
In direct variation, y divided by x is always k.
First pair: 15/3 is 5.
New pair: 40/8 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.