Step by step
The question
Question: find x.
The two lines are parallel.
The alternate angles are 5x − 15 and 3x + 25.
There are four choices.
First we solve it, the normal way.
Then we pick.
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 sits in the same place at its crossing as the bottom angle: an F shape.
So they are equal.
It is also opposite our top angle, where two lines cross.
Opposite angles are equal too.
So the two angles are equal: 5x − 15 = 3x + 25.
2 Get x on one side
There is x on both sides.
We want x on one side only.
The numbers in front of x are 3 and 5.
We remove the smaller one, 3x, so the x part that is left is plus.
So we remove the 3x from the right 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 cancel each other.
They make zero.
5x − 3x = 2x.
Now we have 2x − 15 = 25.
We want x by itself.
First we remove the − 15, so 2x is alone.
Then we remove the times 2.
So after step 2: we took away 3x, and now we have 2x − 15 = 25.
Is x alone now?
Not yet: − 15 is still with x.
How do we remove it?
3 Remove the − 15
To remove − 15, we add 15.
We do it on both sides, so they stay equal.
− 15 and + 15 cancel each other.
They make zero.
25 + 15 = 40.
Now we have 2x = 40.
So after step 3: we added 15, and now we have 2x = 40.
Is x alone now?
Not yet: 2x means 2 times x.
4 Remove the 2
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
They cancel each other.
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 85°.
They are equal, so x = 20 is right!
6 Pick the choice
Our answer is x = 20.
That is choice B.
7 Why not the others
A is 5.
Wrong: took 15 away instead of adding it: 2x = 10.
C is 21.25.
Wrong: made them add up to 180°: alternate angles are equal.
D is 85.
Wrong: that is the angle, not x.
Let's look back.
First: use the angle fact.
Then: get x on one side.
Then: remove the − 15.
Then: remove the 2.
Then: check.
Then: pick the choice.
Last: why not the others.