Step by step
The question
Question: y varies directly with x.
y = 35 when x = 5.
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 = 35 and x = 5.
k times 5 is written 5k.
5k means 5 times k.
We want just k.
To remove times 5, we divide by 5.
We do it on both sides, so they stay equal.
Careful!
k is y divided by x: 35/5.
Not 5/35.
The y number goes on top.
Is k alone now?
Not yet: 5k means 5 times k.
How do we remove it?
35/5 on the left.
5k/5 on the right.
5 divided by 5 is 1.
They cancel each other.
So just k is left.
35 divided by 5 is 7.
So k = 7.
So after step 2: we divided by 5, and now we have k = 7.
3 Write the equation
We put 7 in place of k.
So the equation is y = 7x.
4 Find y
We put in x = 8.
7 times 8 is 56.
So y = 56.
5 Check
Let's check.
In direct variation, y divided by x is always k.
That is how we found k.
First pair: 35/5 is 7.
New pair: 56/8 is 7 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.