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A nice little tutorial for understanding the concept of algebra.

Answer 1

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ALGEBRA, we can say, is a body of rules. They are rules that show how something written one way may be rewritten a different way. For what is a calculation if not transforming one set of symbols into another? In arithmetic, we may replace the symbols '2 + 2' with this symbol '4.' In algebra, we may replace 'a + (−b)' with 'a − b.'

Here are some of the basic rules of algebra:
1· a = a.

(1 times any number does not change it. Therefore 1 is called the identity of multiplication.)

(−1)a = −a.

−(−a) = a.

a + (−b) = a − b.

a − (−b) = a + b.


Associated with these -- and with any rule -- is the rule of symmetry:
If a = b, then b = a.
For one thing, this means that the rules of algebra goes both ways.
Since we may write
(−1)a = −a,

-- that is, in a calculation we may replace (−1)a with −a -- then, on exchanging sides:
−a = (−1)a.
This tell us that we may replace −a with (−1)a.
The rule of symmetry also means that in any equation, we may exchange the sides.
If
15 = 2x + 7,

then we are allowed to write

2x + 7 = 15.
Thus the rules of algebra tell us what we are allowed to write. They tell us what is legal.
Problem 1. Use the rule of symmetry to rewrite each of the following. And note that the symmetric version is also a rule of algebra.
.
Do the problem yourself first!

a) 1· x = x x = 1· x b) (−1)x = −x −x = (−1)x

c) x + 0 = x x = x + 0 d) 10 = 3x + 1 3x + 1 = 10
e) x
y = ax
ay ax
ay = x
y f) x + (−y) = x − y x − y = x + (−y)
g) a
2 + b
2 = a + b
2 a + b
2 = a
2 + b
2
The commutative rules
The order of terms does not matter. We express this in algebra by writing
a + b = b + a
That is called the commutative rule of addition. It will apply to any number of terms.
a + b − c + d = b + d + a − c = −c + a + d + b.
The order does not matter.
Example 1. Apply the commutative rule to p − q.
Solution. The commutative rule for addition is stated for the operation + . Here, though, we have the operation − . But we can write
p − q = p + (−q).

Therefore,

p − q = −q + p.
*
Here is the commutative rule of multiplication:
ab = ba
The order of factors does not matter.
abcd = dbac = cdba.
The rule applies to any number of factors.
What is more, we may associate factors in any way:
(abc)d = b(dac) = (ca)(db).
And so on.
Example 2. Multiply 2x· 3y· 5z.
Solution. The problem means: Multiply the numbers, and rewrite the letters.
2x· 3y· 5z = 2· 3· 5xyz = 30xyz.
It is the style in algebra to write the numerical factor to the left of the literal factors.
Problem 2. Multiply.
a) 3x· 5y = 15xy b) 7p· 6q = 42pq c) 3a· 4b· 5c = 60abc
Problem 3. Rewrite each expression by applying a commutative rule.
a) −p + q = q + (−p) = q − p b) (−1)6 = 6(−1)
c) (x − 2) + (x + 1) = (x + 1) + (x − 2)

d) (x − 2)(x + 1) = (x + 1)(x − 2)
Zero
We have seen the following rule for 0
For any number a:
a + 0 = 0 + a = a
0 added to any number does not change the number. 0 is therefore called the identity of addition.
The inverse of adding
The inverse of an operation undoes that operation.
If we start with 5, for example, and then add 4,
5 + 4,
then to undo that -- to get back to 5 -- we must add −4:
5 + 4 + (−4) = 5 + 0 = 5.
Adding −4 is the inverse of adding 4, and vice-versa. We say that −a is the additive inverse of a. The rule is:
a + (−a) = (−a) + a = 0
A number combined with its inverse gives the identity.
We have seen that that rule is essentially the definition of −a.
Problem 4. Transform each of the following according to a rule of algebra.
a) xyz + 0 = xyz b) 0 + (−q) = −q c) −¼ + 0 = −¼

d) ½ + (−½) = 0 e) −pqr + pqr = 0 f) x + abc −abc = x
g) sin x + cos x + (−cos x) = sin x
The student might think that this is trigonometry, but it is not. It is
g) algebra
Problem 5 . Complete the following.
a) pq + (−pq) = 0 b) z + (−z) = 0 c) −&2$ + &2$ = 0

d) ½x + 0 = ½x e) 0 + (−qr) = −qr f) −π + 0 = −π
g) tan x + cot x + (−cot x) = tan x.

Two rules for equations
Rule 1. If
a = b,

then

a + c = b + c.
This rule means,
We may add the same number to both sides of an equation.
This is the algebraic version of the axiom of arithmetic and geometry:
If equals are added to equals, the sums are equal.
Example 3.
If
x = 2,

then

x + 4 = 6
-- upon adding 4 to both sides.
Example 4.
If
x = 9,

then

x − 4 = 5
-- upon subtracting 4 from both sides.
But the rule is stated in terms of addition. Why may we subtract?
Because subtracting is equivalent to adding the inverse.
a − b = a + (−b)

Subtracting b is the same as "adding" −b.
Therefore, any rule for addition is also a rule for subtraction.
Problem 6.

a) If b) If

x = 2, x = 10,

then then

x + 6 = 8. x − 1 = 9.

c) If d) If

x = −6, x = −2,

then then

x + 2 = −4. x − 3 = −5.

Rule 2. If
a = b,

then

ca = cb.
This rule means,
We may multiply both sides of an equation by the same number.
Example 5. If
2x = 3,

then

10x = ?
Now, what happened to 2x to make it 10x ?
We multiplied it by 5. Therefore, to preserve the equality, we must multiply 3 by 5, also.
10x = 15.
Example 6. If
4x = 14,

then

2x = 7.
In this example, we divided both sides by 2. But the Rule states that we may multiply both sides. Why may we divide?
Because division in algebra is equivalent to multiplication by the reciprocal. In this example, we could say that we multiplied both sides by ½. Therefore, any rule for multiplication is also a rule for division.
Example 7. If
abx = ac,

then

bx = c.
On dividing both sides by a, we say that we have "canceled" the a's. In other words,
If both sides of an equation have a common factor,
then we may "cancel" them.

Problem 7.

a) If b) If

x = 5, x = −7,

then then

2x = 10. −4x = 28.

c) If d) If

3x = 2, −5x = 1,

then then

18x = 12. 25x = −5.

Problem 8. Divide both sides -- mentally.

a) If b) If

3x = 12, 10x = −15,

then then

x = 4. 2x = −3.

c) If d) If

½ax = ½b, pqrx = 8q,

then then

ax = b.
prx = 8.

Problem 9. Changing signs on both sides. Write the line that results from multiplying each side by −1.
a) −x = 5 b) −x = −5 c) −x = 0

x = −5 x= 5 x = −0 = 0.
This problem illustrates the following theorem:
In any equation we may change the signs on both sides.
We will see this when we come to solve equations. For we will see that to "solve" an equation we must isolate x -- not −x -- on the left of the equal sign. And when we come to the distributive rule, we will see that we may change all the signs on both sides.
Problem 10.
a) If x = 9, then −x = −9. b) If x = −9, then −x = 9.


c) If −x = 2, then x = −2. d) If −x = −2 then x = 2.
x is a variable. It is neither positive nor negative. Only numbers are positive or negative. When x takes a value -- positive or negative -- the values of x and −x will have opposite signs. If x takes a positive value, then −x will be negative. But if x takes a negative value, then −x will be positive.
Thus if x = −2, then −x = −(−2) = +2. (Lesson 2.)
(If x = 0, then −x = −0, which we must say is equal to 0. −0 = +0 = 0.)

Answer 2

wow! thats a really long one !

Answer 3

:D

Answer 4

i don't get it.....lol

Answer 5

seems as if u have once written in latex or somewhere and just pasted here.

Answer 6

Appears to be from here http://www.naruto-boards.com/738088/mathematics/

Answer 7

here too: http://www.themathpage.com/alg/rules-of-algebra.htm

Answer 8

^

iGrell is my Naruto-Boards and Naruto-Arena account, also you can find me under Compassionate

@lgbasallote , talked it over with Callisto, MathPage is copied from textbook and is public domain.

Answer 9

so in a way...you copied from a textbook...and still have no citations...?

Answer 10

I do recall posting, "Maybe because most of it are notes from wikipedia and from another site all i did was put them togeter in a helpful way, BUT I do give original and helpful private lessons and I have intuition post of you need an intuition on "something."

Citations aren't required on public domain.

Answer 11

I never took it upon myself to call it originality, so you can't accuse me of fabricating.