Algebra

Algebra

 Learning Outcomes:

After reading this article or post readers will be able to,

• Explain the definition of algebra.
• Know that about sentence and about statement.
• Understand the difference between like and un-like terms.
• Addition and subtraction of algebraic expressions.
• Simplify the given algebraic expressions grouped with brackets.
• Recall basic formulas.
• How to factorize expressions of different types.
• Recognize simultaneous linear equations with one and two variables.
• Different methods to solve simultaneous linear equations.

Algebra:

The word algebra, which is taken from the Arabic word Al Jabar. A famous Muslim mathematician Al khawarzmi wrote a famous book, the name of book is Al Jabar Wal Muqabala in 820 AD, in that book he described the method of solving difficult and complex mathematical problems.

Algebra is an important branch of math which provides us the solution of many complex mathematical problems in an easy way. Especially when we showing or represent a quantity by a symbol without knowing its numerical value.

Relationship between Algebra and Arithmetic:

All of us well aware of natural numbers, which are 1,2,3,4,.... And use of the basic operations like ( +,-, ×,÷ ) in arithmetic.

 In algebra we use letters a,b,c,d,e,.....,z in addition of numbers to generaliz the arithmetic which help us to express a quantity without knowing its numerical value.

 We shall explain it with the help of following example.

1+2 = 3  if x=1, y=2 then z= 3

2+3 =5  if x=2, y = 3 then z=5

3+4=7  if x=3 , y=4 then z=7

....... so on. x+y = z

In above examples x+y = z is giving us a general form which is representing all given Arithmetical statements. So algebra is a general form of the arithmetic.

x + y = z shows the sum of two(2) numbers represented by x and y is equal to the number represented by z.

Sentence:

A group of words that gives us a complete sense is called sentence.

For example, 

• Goats give milk. 
• Birds are flying in the air. 
• He writes very well. 

These are examples of a sentence.

Statement:

A statement is also a sentence that may be true or false.

For example,

a) Banana is a fruit. ( True )

b) Mumbai is in America. ( False )

c) Thank you. (neither true nor false)

Here (a) and (b) are statement but (c) is not. 

Open statement:

A statement from which we cannot decide that, it is true statement or false statement so such type of statement is called open statement. For example,

2 + ∆ = 6 

We try to guess the value of ∆ to make the statement true.

2 + 3 = 6 (false) , 2 + 4 = 6 (true)

Here 4 is the required number which is making the statement true or 4 is satisfying the statement.

Variable and constant:

We learnt in previous post that in algebra a letter of the alphabet (x,y,z etc) is used to represent a number of values. So these letters are called variable, and numbers 0,1,2,3,4,... have definite fixed values are called constants.

Algebraic Expressions:

Such a expressions in which the numbers or variables or both ( variables and numbers) are connected by operational signs (+,-,×,÷) are called algebraic expressions.

For example: 4 , 2x , 5x+3y , a+b , x+y-z etc.

Algebraic terms:

The operational signs plus( + ) and minus (-) separate and connect the parts of an algebraic expression. These parts are called or known as the algebraic terms of the algebraic expression.

For example: in an expression x + 2y + 5 , there are three terms x, 2y , 5

Constant term:

In above expression x + 2y + 5 , the term 5 is a fixed and remains unchanged and only values of x and y vary. So term 5 is called a constant term.

Index or Exponential form:

We know that 2×2×2 = 2³ and 5×5×5×5 = 5⁴. These forms are called index or Exponential forms and can be read as power of 2 and power of 5.

In algebra we can write an index / exponential form as:

y × y = y² or a×a×a = a³

In y², variable y is called base and power 2 is called exponent. We can read it as y raise to the power 2.

Coefficient:

The multiplying factor (mean any Integer) of a variable is called its coefficient. For example in 7y , 7 is the coefficient of y.

Like and un like terms:

The terms of the same kind which are only differ by their coefficients are called like terms. Such terms can be reduced to a single term by subtracting and adding.

 For example,

(1) x + 3x + 4x = 8x , (2) 5xy - 2xy -xy = 2xy

And the terms having different variables or same variables with different exponents are called unlike terms. For example:

(1) x,y,z  (2)  ab, ac ,bc

Addition and subtraction of algebraic expressions:

Please visit my previous post. 

Simplification:

As we know that brackets are used to indicate the order for performing operations. The four (4) kinds of brackets are,

A) _____ is called bar or we called it vinculum.

B) ( ) is called curved brackets or round brackets or parentheses.

C) { } is called braces or curly brackets.

D) [ ] is called square brackets or box brackets.

Sometimes in algebra we cannot Simplify an expression into a single term within the brackets. For example in 3x - ( x + y ) we cannot Simplify the expression ( x + y ). For such situation

1) First expand brackets.

2) Simplify the whole expression as given below,

3x - ( x + y) = 3x - x - y = 2x - y

Hence 2x - y is the simplest form of the above algebraic expression.

-ve ( negative) sign before the brackets means, change the signs of all the terms with in brackets.

 For example. - (x + y - z ) = -x -y +z

Example: [ 5x - { 3b + ( 6a - 2a + b)}]

Solution: 

Example: [ 2a + { c - a + ( a + 2b + c )}]

Solution: 

Evaluation:

The process of finding the numerical value or absolute value of an expression by using numbers in place of variables is called evaluation.

The absolute value or numerical value of an expression varies according to the given value of variables. For example, if a= 2 and b= 1 then the numerical value of a+b is 2+1 = 3 and if a=4 and b=2 then the numerical value of a+b = 4+2 =6.

Example: If a=3 and b = 4 then solve or prove that (a+b)² = a² + 2ab + b²

Solution: 

Basic algebraic formulas:

1) ( x + y )² = x² + y² + 2xy

Example: solve (107)² by using formula.

Solution:

(107)² = (100 + 7 )²

= (100)² + (7)² + 2(100)(7)

= 10000 + 49 +  1400

= 11449 Ans.

2) (x - y )² = x² + y² - 2xy

Example: Evaluate (87)² by using formula.

Solution: 

(87)² = (90 - 3 )²

= (90)² + (3)² - 2(90)(3)

= 8100 + 9 - 540

= 7569 Ans

3) x² - y² = (x - y )(x + y )

Example: evaluate 107 × 93 by using formula.

Solution:

107 × 93 = (100 + 7)(100 - 7)

= (100)² - (7)²

= 10000 - 49

= 9951 Ans.

Factorization:

Factors of an expression are the expressions whose product(multiplication) is the given expression. The process of expressing the given expressions as the product of its factors is called Factorizing or Factorization.

1) Ka + Kb + Kc (type 1)

Example: 2x - 4y + 6z factorize it.

Solution: 2x - 4y + 6z = 2(x -2y + 3z)

Example: x² - xy + xz factorize it.

Solution: x² - xy + xy = x ( x - y - z )

Example: factorize 3x² - 6xy

Solution: 3x² - 6xy = 3x ( x - 2y )

2) ac + ad + bc + bd (type 2)

Example: 3x + cx + 3c + c² factorize it

Solution: 3x + cx + 3c + c²

= ( 3x + cx ) + ( 3c + c² )

= x ( 3 + c ) + c ( 3 + c )

= (x + c ) ( 3 + c )

Example: 2x²y - 2xy + 4y²x - 4y²

Solution:

 2x²y - 2xy + 4y²x - 4y²

= 2y ( x² - x + 2yx - 2y )

= 2y [ x (x - 1 ) + 2y ( x -1 )]

= 2y ( x- 1 )(x + 2y )

3) x² + 2xy + y² and x² - 2xy + y²  (type 3)

Example: 9a² + 30 ab + 25b² factorize

Solution:

9a² + 30 ab + 25b²

= (3a)² + 2(3a)(5b) +(5b)²

= (3a + 5b )²

Example: 16x² - 64x + 64 factorize

Solution:

16x² - 64x + 64

= 16 ( x² -4x + 4 )

= 16  [ (x)² -2(2)(x) + (2)² ]

= 16 ( x - 2 )²

Example: 8x³y + 8x²y² + 2xy³ factorize

Solution:

 8x³y + 8x²y² + 2xy³

= 2xy ( 4x² + 4xy + y² )

= 2xy [ (2x)² + 2(2x)(y) + (y)² ]

= 2xy ( 2x + y )²

4) x² - y² (type 4)

Example: 25x² - 64 factorize it

Solution:

25x² - 64

= (5x)² - (8)²

= ( 5x + 8) ( 5x - 8)

Example: factorize 16y²b - 18bx²

Solution:

16y²b - 81bx²

= b ( 16y² - 81x² )

= b [ (4y)² - (9x)² ]

= b ( 4y + 9x ) ( 4y - 9x )

Example: factorize ( 3x - 5y )² - 49z²

Solution:

( 3x - 5y )² - 49z²

= ( 3x - 5y )² - (7z)²

= ( 3x - 5y + 7z) ( 3x - 5y - 7z )

Example: By using formula evaluate (677)² - (323)²

Solution:

(677)² - (323)²

= (677 + 323) ( 677 - 323)

= 1000 × 354

= 354000

5) x² + 2xy + y² - z² and x² - 2xy + y² - z² (type 5)

Example: a² - 2ab + b² - 4c²

Solution:

( a² - 2ab + b² ) - 4c²

= ( a - b )² - (2c)²

= ( a - b - 2c ) ( a- b +2c )

Example: 4a² + 4ab +b² - 9c²

Solution:

 4a² + 4ab +b² - 9c²

= (2a)² + 2(2a)(b) + (b)² - 9c²

= (2a+b)² - (3c)²

= ( 2a + b - 3c)( 2a + b + 3c )

Cubic formulas:

1) ( x + y )³ = x³ + y³ + 3xy(x + y )

Example: solve ( 3a + 4b )³

Solution: ( 3a + 4b )³

= (3a)³ + (4b)³ + 3(3a)(4b)(3a + 4b)

= 27a³ + 64b³ + 36ab(3a + 4b)

= 27a³ + 64b³ + 108a²b + 144ab²

2) (x - y )³ = x³ - y³ - 3xy ( x - y )

Example: ( 2a - 3b )³

Solution:

( 2a - 3b )³

= (2a)³ - (3b)³ - (3)(2a)(3b)(2a - 3b )by formula

= 8a³ - 27b³ - 18ab (2a - 3b )

= 8a³ - 27b³ - 36a²b + 54ab²

Simultaneous linear equations:

If there are two or more linear equations consisting of same set of variables are satisfied simultaneously by the same values of the variables then these equations are called simultaneous linear equations.

Recognizing simultaneous linear equations in one(1) and two(2) variables:

As we know that a linear equation is an algebraic equation in which each term is either a variable or constant or the product of variable or a constant. The standard form of linear equation consisting of one (1) variable is,

ax + b ,  ∀ a,b ∈ R

Similarly a linear equation in two variables is of the form ax + by = c where a,b and c are constants. Two (2) linear equations considered together from a system of linear equations. For example  x + y = 3 and x - y = 2 is a system of two linear equations with two(2) variables x and y. This system of two (2) linear equations is known as the simplest form of linear system which can be written in the general form as,

a₁x + b₁y = c₁

a₂x + b₂y = c₂

Solution of simultaneous linear 

equations:

There are many methods to solve simultaneous linear equations that is

1) Method of equating the coefficients.

2) Method of elimination by substituting.

3) Method of cross Multiplication.

Have a nice day.