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Tuesday, December 11, 2007

Chapter 4: Linear Equations (I)

4.1 Linear Equations In One Unknown

1. A linear term is a term with one unknow and the power of the unknown is one.
Example:
5m, 10n, ½p → Linear terms
8x², 3ab, 6y ³z → Non-linear terms
2. A linear algebraic expression is formed when two or more terms with the power of one are combined by addition or subtraction or both.
Example:
3x-4y → linear expression
4m+5n → linear expression
a²+3 → not a linear expression
3. A linear equation involves only expressions where the power of the unknowns is one
Example:
4m + 7n = 10 → linear equation
9a² + 3 = 8 → not a linear equation
4. A linear equation in one unknown is an equation which shows the relationship between numbers and a linear term with one unknown.
Example:
2k + 3 = 6 → linear equation in one unknown
4x + 5y = 8 → not a linear equation in one because it consists of two unknowns, x and y.

4.2 Solutions of Linear Equations in One Unknown

1. Solving linear equation is a process of finding the value of an unknown which satisfies the equation.
2. the solutions of the equations are also known as the roots of the equations.
3. A linear equation in one unknown has only one root.
4. We can determining the solution of a linear equation by trial and improvement method, but not all the equations can be easily solved by this method.
5. There are 4 methods to solve linear equations in one unknown, depending on the form of the equations.
Example:
(i) Using subtraction (ii) Using addition
x + 8 = 14
x + 8 -8 = 14 - 8
x = 6

(ii) Using addition
x - 3 = 7
x - 3 + 3 = 7 + 3
x = 10

(iii) Using division (iv) Using multiplication
5x = 15 x/5 = 6
5x ÷ 5 = 15 ÷ 5 x/5 x 5 = 6 x 5
x = 3 x = 30

(iv) Using multiplication
x/5 = 6
x/5 x 5 = 6 x 5
x = 30

Sunday, November 11, 2007

Chapter 3: Algebraic Expressions (II)

3.1 Algebraic Terms In Two Or More Unknowns

1. An algebraic term in two or more unknowns is a product of the unknowns with a number.
Example:
5xy = 5 x x x y
Therefore, 5xy is the product.

2. The term implies that the unknown m is multiplied a times by itself. This is known as repetitive multiplication of an unknown.
Example:
k³ = k x k x k
k is multiplied 3 times.

3. Normally, the product of the unknowns number is expressed without the mutiplication
symbol.
6pqr = 6 x p x q x r

4. The coefficient of an unknown in an algebraic term is the factor of the algebraic term.
Example:
2xy
2 is the coefficient of xy

3.2 Multiplication and Division of Two or More Algebraic Terms

1. The product of two algebraic terms can be found by multiplying the numbers and the same
unknowns separately.
Example 1:
3mn x 4mn²p = 12m²n³p

2. The quotient of two algebraic terms can be found using the cancellation method.
Example 2:
40xy²z ÷ 8xy = 5yz
we can cancel x and y

3.3 Concept of Algebraic Expressions

1. Algebraic expression can be written using letter symbols.

2. In algebra, letters are used to represent unknowns and numbers.
Example 1:
Assume a number, we can assume it by x.
Then we multiplied this number by 8 → 8x
After that, we add 10 from the term → 8x + 10
So, now we can express the above given as 8x + 10.

3. An algebraic expression consists of two or more algebraic terms combined by addition or
subtraction or both.

4. Combinations of algebraic terms in two or more unknowns will produce algebraic expression
in two or more unknowns.
Example 2:
Algebraic expression in two unknowns → 3mn - mn; 10xy + 2xy
Algebraic expression in more than two unknowns → 2ab - 3mn + 4xy; a² + 2ab +cd³.

5. Only expression with similar term can be adding or subtracting the coefficient of the
unknowns

6. To simplify an expression that consists of a few like and unlike terms:
(i) rearrange and group the like terms
(ii) then, add or substract the coefficient of the unknowns
Example 3:
5mn + 4mn = 9mn
* This term can be added because the unknown is same (like terms)

Example 4:
6mn-4pq = 6mn - 4pq
* This term can not be subtracted because the unknowns is different (unlike terms)

7. The expression can be evaluated when the unknowns is substituted with the given number.
Example 5:
Find the value of 3mn² + 2p²q - mp; given that m=1, n=2, p=-1 and q=-2
Firstly, we subtitute all the unknowns into the expression, its becomes:
3(1)(2)² + 2(-1)²(-1) - (1)(-1)
=12 - 2 + 1
= 11 * The answer is 11 after the calculation

3.4 Computations Involving Algebraic Expressions

1. Multiplying algebraic expression by a number
To simplify the product of an algebraic expression with a number, each term in the expression
must be multiplied by the number:
Example 1:
Simplify 8(3mn - 4xy)
= 24mn - 32xy

2. Dividing algebraic expession by a number.
To simplify the quotient of an algebraic expression with a number, each term in the
expression must be divided by the number:
Example 2:
Simplify (4ab² - 8c³d + 12ef) ÷ 2
= 4ab²/2 - 8c³d/2 + 12ef
= 2ab² - 4c³d + 6ef

3. Adding two algebraic expression.
To find the sum of two algebraic expression, we can simplify the algebraic expression using
the following method:
→Remove brackets and groups the like terms.
→Find the sum or difference of the like terms.
Example 3:
Simplify (2mn + 4p²) + (3mn - 5p²)
= 2mn + 4p² + 3mn - 5p²
= 2mn + 3mn + 4p² - 5p²
= 5mn - p²

4. Subtracting two algebraic expressions.
To find the difference between two algebraic expressions, we can simplify the algebraic
expressions, we can simplify the algebraic expressions using the following methods:
→Remove brackets and change the symbols of the terms in the subtracted algebraic
expression. Then, group the like terms
→Find the sum or difference of like terms.
Example 4:
Simplify (4m²n - 5pq) - (mn - 2pq - 6m²n)
= 4m²n - 5pq - mn + 2pq + 6m²n
= 4m²n + 6m²n - 5pq + 2pq - mn
= 10m²n - 3pq - mn


Tutorials (2):

1. (-4a²b) ÷ 3b²c² x 9ac 2. 3pq x 6rp²q

3. Given that p=-3 and q = 2, 4. 3mx² - 8n²y + (-4mx²) - n²y
evaluate p² + (-pq²) - p²q²

5. -15(-x²yz² + 2pq² - 3mn) = 6. (p³q x -9pq) ÷ 3p



Sunday, November 4, 2007

Chapter 2: Squares, Square Roots, Cubes and Cube Roots

2.1 Squares of Numbers

The square of a number is the product of a number multiplied by itself.

Example:
The square of 10 is 10 x 10 = 100
The square of 0.5 is 0.5 x 0.5 = 0.25
The square of 1/4 is 1/4 x 1/4 = 1/16

The square of a number can be written using the square notation.

Example:

6 x 6 can be written as 6².

0.3 x 0.3 can be written as 0.3².

(-8) x (-8) can be written as (-8)².

*Remember: The square of any numbers always is positive!

such as (-2)² = -2 x -2 = 4; still remember, (-) x (-) = (+)

2.2 Square Roots

1. The symbol for square root is √¯¯.

Example 1:

What is the square root of 36?

Solution:

The square root of 36 can be written as √36. (We know 36 is the product of 6 x 6)

So, the answer for this question is 6!

Example 2:

What is the square root of 9/16?

We can write this as √(9/16). (We know that 9/16 is also equal to (3x3/4x4).

So the answer for this question is 3/4!

* Remember: Square root of any numbers must be positive. If negative, that answer is

undefined!

2.3 Cubes Of Numbers

The cube of a number is the product of the number multiplied by itself twice.

The cube of a number can be written using the cube symbol or notation ³.

Example 1:

The cube of 3 can be written as 3³ = 3 x 3 x 3.

The cube of -2 can be written as (-2) ³ = (-2) x (-2) x (-2).

The cube of 1/5 can be written as (1/5)³ = (1/5) x (1/5) x (1/5).

Example 2:

Calculate the value of the cube of four.

4³ = 4 x 4 x 4 = 64

Calculate the value for 0.5³

0.5³ = 0.5 x 0.5 x 0.5 = 0.125

2.4 Cube Roots

The cube root of a number is a number which, when multiplied itself twice. It can be written as the symbol as ³√¯¯.

Example 1:

The cube root for 125 is 5 because 125 = 5 x 5 x 5

The cube root of -8 is -2 because -8 = (-2) x (-2) x (-2)