While I agree with what the previous responders have said, I've had some experience with this type of problem.
You'll notice that the topic was "Hands On Equations." This type of exercise is usually done using manipulatives, such as plastic chips. There is one type of chip (maybe a white square) used to represent each x. Another type of chip (perhaps a black square) is used to represent -x (or the "x with the line through it"). A white chip and a black chip, taken together, represent 0. A white chip and black chip can be removed together from one side of the equation without changing anything (not surprising, that.) Another type of chip, maybe a red circle, would be used to represent 1 unit.
The given problem,
2x + (-x) + 2x = 10 + x
could be represented with manipulatives this way (often the left and right sides of the equation are shown as the pans of a balance scale, and the equals sign is the balance point for the two pans):
2 white squares + 1 black square + 2 white squares = 10 red circles + 1 white square
The "goal" is to get 1 white chip by itself on one side of the balance scale, and some number of red chips on the other side.
On the left side, 1 of the white squares and the black square can be removed since they "cancel each other out." Now we will have
1 white square + 2 white squares = 10 red circles + 1 white square
We can now take 1 white square off each side without disturbing the balance:
2 white squares = 10 red circles
Now, if we divide what is on each side of the equals sign by 2, things should still balance:
1 white square = 5 red circles
x = 5
And since a black square "cancels out" a white square, 1 black square, or -x, is -5.
I did this in my pre-algebra classes because it was required, not because I thought it was of any great value. The smarter kids caught on right away, and easily moved using symbols like x and -x.....perhaps it has a benefit for students who learn best using a more "physical" technique. Seems to me this all came about at the same time that there was a lot of emphasis on different learning styles.
I hope this explains a bit what was going on.....