Step by step
The question
Question: y varies directly with x.
y = 14 when x = 4.
Find y when x = 6.
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 = 14 and x = 4.
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 x: 14/4.
Not 4/14.
The y number goes on top.
14/4 on the left.
4k/4 on the right.
4 divided by 4 is 1.
So just k is left.
We can make 14/4 simpler.
It is the same as 7/2.
So k = 7/2.
3 Write the equation
We put 7/2 in place of k.
So the equation is y = 7/2 times x.
4 Find y
We put in x = 6.
7/2 times 6 is 21.
So y = 21.
5 Check
Let's check.
In direct variation, y divided by x is always k.
First pair: 14/4 is 7/2.
New pair: 21/6 is 7/2 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.