Step by step
The question
Question: find x.
AB is a diameter.
The angles at A and B are 3x and 2x + 15.
1 Use the circle rule
AB goes through the center O.
It is a diameter.
It cuts the circle in half: two semicircles.
At the center, A and B make a straight line: 180°.
The angle at C, on the edge, is half of it: 90°.
A right angle.
2 Use the angle fact
The three angles in a triangle always add up to 180°.
So we add them all and make it 180: 3x + 2x + 15 + 90 = 180.
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.
3x + 2x = 5x.
15 + 90 = 105.
Now we have 5x + 105 = 180.
We want x by itself.
Two things are stuck to x: times 5 and + 105.
The + 105 is on the outside, so it comes off first.
Then the times 5.
4 Remove the + 105
To remove + 105, we take away 105.
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.
+ 105 and − 105 make zero.
Cross them out.
180 − 105 = 75.
Now we have 5x = 75.
5 Remove the 5
5x means 5 times x.
We want just x.
To remove times 5, we divide by 5.
We do it on both sides, so they stay equal.
5 divided by 5 is 1.
So just x is left.
75 divided by 5 is 15.
So x = 15!
6 Check
Let's check: put x = 15 in each angle.
Now add all the angles.
They add up to 180°.
So x = 15 is right!
Let's look back.
First: use the circle rule.
Then: use the angle fact.
Then: group like terms.
Then: remove the + 105.
Then: remove the 5.
Last: check.