Thank you, this was really helpful
I like the fact about algebra that you can do anything you like to attempt to figure out the answer by doing whatever you like to the equation as long as you do it to both sides of the equals sign (=), just pull something out of thin air and do it to both sides and you are safe. Undo it and try another balanced attempt and see what you come up with next try. Algebra is a balancing act of life, without algebra, there is no life, there is only chaos.
^
This.
The name of algebra originally meant "solving by balancing and completion". It is a method of trying to "undo" what was "done" to an unknown quantity so that it becomes known. A typical example:
2x + 5 = 11
To get 11, we multiplied x by 2, and added 5. So now we work backwards to "re-discover" x. How do we undo "adding 5?" We subtract 5, and to preserve equality, we subtract 5 from BOTH 2x + 5 and 11:
(2x + 5) - 5 = 11 - 5
11 - 5 is known to be 6 (this is easy enough to do by "counting backwards on one's fingers"). So now we have:
(2x + 5) - 5 = 6
It is also known that we can re-arrange the parentheses like so:
2x + (5 - 5) = 6.
(This is, formally, called associativity of addition:
(2x + 5) - 5 = (2x + 5) + -5 = 2x + (5 + -5) = 2x + (5 - 5), in excruciatingly painful detail).
It is likewise also known that 5 - 5 = 0. So this brings us to:
2x + 0 = 6.
Since adding 0 is the same as "not adding anything" (this is what we MEAN when we say: "0 is an additive identity"), we now have:
2x = 6
Already we have made a great deal of progress, We now have just 2 terms, instead of 3, with a much simpler form of equation. To undo "multiplication by 2" we DIVIDE by 2:
(2x)/2 = 6/2.
Since 6 = 2*3, 6/2 = (2*3)/2 = (2/2)*(3/1) = (1/1)*(3/1) = 3/1 = 3. So now we have:
(2x)/2 = 3.
Finally, we have:
(2x)/2 = (2x)/(2*1) = (2/2)*(x/1) = (1/1)*(x/1) = x/1 = x.
This tells us that:
x = 3, and we have solved the mystery.