Algebra Intro

Algebra 100 Intro

Algebra is a scary word, but really, it's just a fancy way of complicating the simple fact that a mathematical equation is unknown or incomplete.
For example, we know that:

1 + 1 = 2.

So therefore, if x = 1, then x + x = 2
because x is 1 and 1 + 1 = 2.
That's easy.
Let's try that again.
2 + 1 = 3
If a variable named y is equal to 1,
then 2 + y = 3
because we defined the variable y to equal 1.
Now what if we don't know one of the numbers, perhaps the 2. In that case, when we write the above equation, instead of a 2, we place a variable[D: a character representing a numeric value or number] in its place.
Let's try it again.
The variable p = 3
And the variable q is unknown
But we know the following equation is true:
   p + p =  q
Then q must equal 6, right?
Because p = 3,
Then p + p = 3 + 3 = 6
Therefore, q = 6.

Variables

A variable is anything representing something else in mathematics or science. Computers, physics, engineering, and many more science topics all rely on variables. A variable acts as a symbol of a number we do not yet know for sure. So now, with a variable of X in place of the 2, the equation would be written as follows:

1 + 1 = X

Or we could say:

          1 + x = 2

You can decide what character you use as a variable, but keep in mind the following:
If you draw or write a symbol not available on a keyboard, then when you want to type the equation and your brain is working already on making sure you copy it right from paper, then remembering that cool symbol you drew before is now another thing to keep track of and could result in errors down the line. Meaning, keep it simple or as simple as possible for the sake of ease of use or understanding.

Tangent

As equations get more complicated, small mistakes in the beginning could grow into big mistakes later. Avoiding complications will reduce errors.

Variable Symbols

Also, using symbols such as a dollar sign ('$') or something you found on your new mobile device, perhaps whatever this symbol is: ¥, could already be used as a reserved symbol representing something else; for example, π = 3.14 approximately.
Let me explain. While you are free to choose any character or symbol at your discretion, you are not the first person to learn Algebra. Many others have learned the hard way that if everyone uses different variables to represent the same thing, then when someone else looks at your version of the same equation, precious time will be wasted translating your variable choices to the ones they will have understood.
So, what people did with their fancy public school systems served a la suburbia was agree on certain variables being standards for certain things.
One day, I'll link to a comprehensive spreadsheet of all industry standard variables so we can all enjoy that saved time on better moments, but for now, here's some guidelines.

Variable Guidelines

Typical math numerals are italicized letters, usually lower case, but sometimes upper case, such as if you choose X, because a lower case X could be confused with a multiplication symbol. So of course, lower case x is the standard for multiplying the two numbers on either side of it. The asterisk ('*') is also used for multiplying as the standard for computer programmers, and has replaced the letter X on graphic user interfaces ('GUI').
Another example is the lower case i, which is reserved in advanced mathematical theorems for the imaginary number. Rest assured, any mathematics dependent on this standard symbol will:
1) Never be useful in your future, and
2) Not be taught in Papoose's mathematics.
The first equation we were working on is simple.
If:

1 + 1 = X

And because we know that 1 and another 1 gives us 2,
Then we can logically deduce that in order for the equation to be true, X must equal 2;
Otherwise, the equation would not be true and the equal sign would have to be replaced with a greater than or less than symbol for any other value of X other than 2.
    1 + 1 < 3

    1 + 1 > 1.5
So, in order for an equation to be true, you must solve for the unknown variable while maintaining the equilibrium of the equation. That means any changes must be made to both sides of an equation and in equal proportion. So if you add 1 to the left side of an equal sign, then you must add 1 and only 1 to the right side of an equal sign.
For example, 1 + 1 = X
So therefore, if we add -1 to both sides, we get:
1 + 1 - 1 = X - 1
Then a one and a negative one on the left side cancel each other out by equaling zero, so
1 + 0  = X - 1
and then
1 = X - 1
To return to the original equation, we add one to both sides again as follows:
1 + 1 = X - 1 + 1
So then
1 + 1 = X + ( -1 + 1)
  2     =  X  + 0
  2     =    X
So then we see once again that X = 2 in that equation. In other equations, the variable X may have different values, or you may use a different variable other than X.