Monday, March 4, 2013

What is an equation?

I am not trying to make a definition of equation in this moment, but I will try to open your eyes in this sense when you solve word problems:

AN EQUATION IS A CONDITION

When we solve problems, we receive many data and CONDITIONS.

So, if we see an equation as a condition, we can easily find the way to the right answer:

Try to convert 
EACH CONDITION 
in 
ONE EQUATION

This is very important: if we receive one condition, there must be an equation to be used to solve the problem.

But if we receive two or more conditions, we must consider seriously the chance to look each condition as an equation, so we have to look for so many equations as conditions we find.

We are going to give you some examples of this soon.

The DIMENSIONAL TRICK for solving algebraic equations. Theory

We have tried to solve many problems that we had no idea how to deal with. We have some data, but we don´t know what to do with them. I present you the dimensional trick useful in this type of problems.

First we have to state as follow. When we use an equation, we have a first member equal to a second member, so both sides of the equation must be referred to the same kind of "things".

You must notice that in the first member we can see an addition or a subtraction. This is obviuos. If I calculate "oranges + oranges", the only posible answer must be "oranges" (not apples).

No matter how many oranges we are dealing with. This is not the point. We are focus now only in the type of things that are involved in equations.

Other examples of the same idea:
I remark: always referred to an addition or a subtraction. Obviously, if we interchange first and second member, the concept is the same.

So, in contrary sense, we must say that:

Of course, my dear friend, if I calculate "oranges + stones" the result can´t be "oranges" nor "stones". (Again, no matter how many oranges or stones).

Another step forward. Now we are thinking about flow formula. Flow is the amount of something that flows through any kind of place in certain time.


As we can state easily, if flow is amount over time, then time must be amount over flow. Now we are going to use this "new" view of time as a fraction, so:
If we add "amount over flow" (I mean, time) and "amount over flow" (I mean, time too), then the result must be time.

If we use another kind of matters, the result is the same:

So:

"Velocity · time" (space) plus "velocity · time" must be an space. No other result is posible.

Thinking about this idea, we have to apply it to solve a realistic algebraic problem in a second part of this article very soon.

Saturday, February 16, 2013

What is the difference between identity and equation?

An identity is an expresion that is true for any value of the letters:

x + 1 = 2x + 2              (is true for any value of x)
(a+b)·(a+b) = a2 + 2ab + b2            (is true for any value of a and b)

An equation is valid only for a small number of values of the letters:

x + 3 = 5                    (is true only for x = 2)
x2 - 1 = 0                  (is true only for x=1 and x=-1)


Factorization techniques for algebraic expresions

Main factorization techniques for polinomials or algebraic expresions:

- First of all, you can look for a common factor in all summands. If it exists, you can pull it out with the distributive property.

- You can use identities like the square of an addition, the square of a subtraction, and addition by subtraction.

- You can factorize a polinomial using ruffini rule. By this method, you will find the solutions of that equation, and this will be easy to make it in a multiplication way.

- If the equation is in the second degree, it is not neccesary to use Ruffini rule, because we can find the solutions using the main formula for that kind of equations.

What does "factorize" mean?

When we learn maths, we have receive the problem to factorize some expresions or numbers, but what does it means?

Factorize is the process by which we convert anything (a number or algebraic expresion) in a multiplication.

So, if we must factorize a number, we will transform that number in a multiplication of primes numbers:

factorization of 24 = 2·3·4

And factorization of an algebraic expresion, is another expresion transformed in a multiplication instead of an addition:

x2 + 2x + 1 = (x+1)·(x+1)

Remember: factor is a part of a multiplication, in the same way that a summand is a part of an addition.

How to know if a fraction is irreducible?

The best way to know is a fraction is irreducible is:

- Factorize both numerator and denominator.

- If there is no factor equal in numerator and denominator, the fraction is irreducible.

- If there is any factor equal in both numerator and denominator, just erase the same factor in both parts and you will find de irreducible fraction.

Another way to say this is that "numerator and denominator are mutual primes" (they don´t have any factor in common).

How to find quickly an irreducible fraction?

If you have a fraction to simplify, it is normal to try some numbers dividing for numerator and denominator in order to find the irreducible fraction.
The best and quick way to proceed to avoid loss of time is:

- Calculate the greatest common divisor between numerator and denominator.

- Now divide numerator and denominator by this number. The fraction you have obtain is the irreducible one.