Step by step
The question
Question: find x.
The two lines are parallel.
The alternate angles are 3x + 10 and 5x − 30.
1 Use the angle fact
The arrows show the two lines are parallel.
Parallel lines go the same way.
They never meet.
A third line crosses both of them.
It is called a transversal.
Our two angles are between the parallel lines, on opposite sides of the transversal.
They are called alternate angles.
Together they make a Z shape.
Alternate angles are equal.
Why?
Look at this angle at the top.
It is in the same place as the bottom angle, so they are equal.
It is also opposite our top angle, so they are equal too.
So the two angles are equal: 3x + 10 = 5x − 30.
2 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the 3x from the left side.
To remove 3x, we take away 3x.
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.
3x and − 3x make zero.
Cross them out.
5x − 3x = 2x.
Now we have 10 = 2x − 30.
Let's swap the two sides, so x is on the left.
We want x by itself.
Two things are stuck to x: times 2 and − 30.
The − 30 is on the outside, so it comes off first.
Then the times 2.
3 Remove the − 30
To remove − 30, we add 30.
We do it on both sides, so they stay equal.
− 30 and + 30 make zero.
Cross them out.
10 + 30 = 40.
Now we have 2x = 40.
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.
40 divided by 2 is 20.
So x = 20!
5 Check
Let's check: put x = 20 in each angle.
Both angles are 70°.
They are equal, so x = 20 is right!
Let's look back.
First: use the angle fact.
Then: get x on one side.
Then: remove the − 30.
Then: remove the 2.
Last: check.