Step by step
The question
Question: 28 students: 14 play chess, 10 play cards, 9 play neither.
How many play both?
1 Draw the Venn diagram
We don't know the middle.
We call it x.
Play chess: 14 in all.
x of them are in the middle.
So 14 − x play only chess.
In the same way, 10 − x play only cards.
9 play neither: outside the circles.
2 Write the equation
All four parts together make 28.
Careful!
33 is more than 28.
The ones in the middle were counted twice: once in each circle.
Put the parts in: (14 − x) + x + (10 − x) + 9 = 28.
The brackets are only to show the parts: a plus in front, so they can just go.
3 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the x terms go together.
Plain numbers go together.
14 + 10 + 9 = 33.
−x + x − x = −x.
Now we have 33 − x = 28.
4 Remove the 33
We want x by itself.
The 33 is in the way.
To remove 33, we take away 33.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
33 and − 33 make zero.
Cross them out.
28 − 33 = −5.
Now we have −x = −5.
5 Remove the −1
−x means −1 times x.
We want just x.
To remove times −1, we divide by −1.
We do it on both sides, so they stay equal.
−1 divided by −1 is 1.
So just x is left.
−5 divided by −1 is 5.
So x = 5!
6 Check
Let's check: put 5 in place of x.
We get 28, the same as the right side.
So x = 5 is right!
So 5 play both.
In the picture: 9 only chess, 5 both, 5 only cards, 9 neither.
Together 28.
So it fits!
Let's look back.
First: draw the Venn diagram.
Then: write the equation.
Then: group like terms.
Then: remove the 33.
Then: remove the −1.
Last: check.