how to do math
how do you do algebra
Well, alaina, this is a very general question, and people spend whole careers on such questions.
At a fundamental level, Algebra is the study of variables - unknown quantities that we would like to solve for.
Here's an example: If you have a certain number of apples, and a friend of yours has 5 more apples than you do, and you two have 15 apples altogether, how many do you have, and how many does your friend have? We represent the unknown quantity "how many do you have" as a variable, x. Since your friend has 5 more than you do, she has x+5. Then, since you have 15 between you, we can say:
x + (x+5) = 15.
The two x's combine:
2x + 5 = 15
Then we subtract 5 from both sides:
2x = 10
and then divide both sides by 2:
x = 5.
Therefore, you have 5 apples, your friend has 10 apples (5 more than what you have) and between you, you have 15 apples, as required by the question.
Algebra was invented to handle unknown quantities more easily. For example, in Euclid's time, the modern distributive property was written something like this proposition from Chapter 2 of Euclid's Elements:
"If there are two straight-lines, and one of them is cut into any number of pieces whatsoever, then the rectangle contained by the two straight-lines is equal to the (sum of the) rectangles contained by the uncut (straight-line), and every one of the pieces (of the cut straight-line)".
Notice how complicated this sentence is. But that is the way statements in math had to be written before algebra.
Once algebra was invented, the statement above can be simply be written as:
a(b+c+d+...) = ab+ac+ad+...
Which is the familiar distributive property.
By having the convenient notation we call algebra, mathematics becomes easier to do.
Exercise for you:
Given 4x - 8 = 0.
And the step-by-step solution:
I. 4x = 8
II. x = 8/4
III. x = 2
Use only english sentences and no algebraic notation to state the above problem and solution and then compare. Use the words "the unknown quantity" in place of the letter x. Which of the two do you prefer?
In elementary school, we learn various ways to solve problems, but mostly we are learning the basics of computation or arithmetic. I like to think of algebra as being a detective to find a mystery number, which instead of having a box or shaded area, now uses a letter.
So, what you used to see as 13 + ___ = 18, you may now see written as: 13 + n = 18.
Because we are "pretending" to be detectives, we have to take notes on our logic of finding "clues."
So instead of saying, "Oh, that's easy. I know it is 5!" I must now show the logic by "isolating" the variable. Think about it as getting the variable alone so you can make an accusation like in the game Clue.
Here's how this problem would work:
The = is the balance of the equation and everything needs to stay balanced, so what I do to one side, I MUST do to the other side.
13 + n = 18 I will get n all by itself by taking the 13 off the left side of the balance.
-13 = -13 I always do the opposite operation, so if 13 is positive, I will subtract. If the
number is negative, I will add. If it is written like 4n, it means 4 times n and I will
divide to get rid of the number. If it is written like a fraction ( n/3), it means divide,
so I will multiply the number on the bottom to cancel it out.
n = 5 13 - 13 is 0, so it is gone and leaves me with my "mystery number" which is 5!!
Yes, I know that is simple, but the problems get gradually more and more complex, so if you understand this basic idea that we are being logical detectives and not just "knowing" the answer, we can solve MANY very difficult problems step-by-step with ease and not guess.
I hope that helps explain the difference between how to do "math" and how to do algebra.