Step by step
The question
Question: find x.
PA and PB are tangents from P to the circle with center O.
PA = 3x − 1 and PB = x + 7.
1 Use the tangent rule
PA and PB touch the circle at A and at B.
Both are tangents from the point P.
Join the center O to A and to B.
OA and OB are both radii, so they are equal.
Each radius meets its tangent at a right angle.
OP is a side of both triangles.
Right angle, hypotenuse OP, side: the two triangles are congruent, by R H S.
So PA = PB.
Two tangents from one point are always equal: 3x − 1 = x + 7.
Careful!
The two tangents are the same length.
So one equals the other.
2 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the x from the right side.
To remove x, we take away x.
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.
x and − x make zero.
Cross them out.
3x − x = 2x.
Now we have 2x − 1 = 7.
We want x by itself.
Two things are stuck to x: times 2 and − 1.
The − 1 is on the outside, so it comes off first.
Then the times 2.
3 Remove the − 1
To remove − 1, we add 1.
We do it on both sides, so they stay equal.
− 1 and + 1 make zero.
Cross them out.
7 + 1 = 8.
Now we have 2x = 8.
4 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
8 divided by 2 is 4.
So x = 4!
5 Check
Let's check: put x = 4 in each tangent.
Both are 11.
So x = 4 is right!
Let's look back.
First: use the tangent rule.
Then: get x on one side.
Then: remove the − 1.
Then: remove the 2.
Last: check.