Step by step
The question
Question: 30 students: 18 play football, 15 play tennis, 5 play neither.
How many play both?
1 Draw the Venn diagram
We don't know the middle.
We call it x.
Play football: 18 in all.
x of them are in the middle.
So 18 − x play only football.
In the same way, 15 − x play only tennis.
5 play neither: outside the circles.
2 Write the equation
All four parts together make 30.
Careful!
38 is more than 30.
The ones in the middle were counted twice: once in each circle.
Put the parts in: (18 − x) + x + (15 − x) + 5 = 30.
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.
18 + 15 + 5 = 38.
−x + x − x = −x.
Now we have 38 − x = 30.
4 Remove the 38
We want x by itself.
The 38 is in the way.
To remove 38, we take away 38.
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.
38 and − 38 make zero.
Cross them out.
30 − 38 = −8.
Now we have −x = −8.
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.
−8 divided by −1 is 8.
So x = 8!
6 Check
Let's check: put 8 in place of x.
We get 30, the same as the right side.
So x = 8 is right!
So 8 play both.
In the picture: 10 only football, 8 both, 7 only tennis, 5 neither.
Together 30.
So it fits!
Let's look back.
First: draw the Venn diagram.
Then: write the equation.
Then: group like terms.
Then: remove the 38.
Then: remove the −1.
Last: check.