Step by step
The question
Question: if 9x + 6y = 12, what is 3x + 2y?
We want 3x + 2y.
Let's look for it inside the equation.
1 Compare the two
Look at 9x + 6y and 3x + 2y.
We compare them part by part.
9 divided by 3 is 3.
So 9x is 3 times 3x.
6 divided by 2 is 3.
So 6y is 3 times 2y.
Every part of 9x + 6y is 3 times a part of 3x + 2y.
2 Take out the 3
So we can write each part as 3 times.
3 times each part is the same as 3 times the whole bracket: 3(3x + 2y).
Opening the bracket gives 9x + 6y back.
3 Remove the 3
3(3x + 2y) means 3 times the bracket.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just the bracket is left.
12 divided by 3 is 4.
So 3x + 2y = 4!
Careful!
Both sides are divided by 3.
12 becomes 4.
4 Check
We can't find x and y on their own: many pairs work.
Let's check with one pair.
We pick x = 0, so that y is a small whole number.
With y = 2, 9x + 6y is 9 × 0 + 6 × 2 = 0 + 12 = 12.
It fits the question.
And 3x + 2y is 3 × 0 + 2 × 2 = 0 + 4 = 4.
The same answer.
It works!
Let's look back.
First: compare the two.
Then: take out the 3.
Then: remove the 3.
Last: check.