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Algebra
Concept of algebra originated as a problem solving technique in which unknown numbers (quantities to be found) are denoted by letters and together with known numbers are used to form expressions and equations that can be solved for the unknowns.Even complex mathematical statements can be easily put into equation using algebraic expressions.
It is said that algebra as a branch of Mathematics began about 1550 BC, i.e.more than 3500 years ago, when people in Egypt started using symbols to denote unknown numbers.
Diophantus ( lived around 250 AD) of Alexandria, sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica, many of which are now lost. These texts deal with solving algebraic equations.
The word ‘algebra’ is derived from the title of the book,‘Aljebar w’al almugabalah’, written about 825 AD by an Arab mathematician, Mohammed Ibn Al Khowarizmi of Baghdad.
Variable
The unknown quantities in mathematical expressions denoted by letters are called variables. (x,y etc.)The number of angles in a triangle = 3
The number of angles in two triangles = 3×2 = 6
The number of angles in three triangle = 3×3 = 9
So, we can say that the number of angles in any n triangles = 3×n
Here, 3, 6 and 9 are fixed values, such numbers are called constants but n can take any value, such numbers are called variable.
Another example-
Ram is three older than Shyam,
When Shyam is two years old Ram's age = 2+3 = 5 years
When Shyam is five years old Ram's age = 5+3 = 8 years
So, we can say that when Shyam is x years old Ram's age = (x +3) years
Here, x is a unknown number and hence a variable.
➢ Variable are used to write formulas and rules in general way applicable to any number.
➢ They are used to form expressions and equations .
Use of variable in common rules and formulas
Some Rules from geometry
Formula for Perimeter of a squareA square has four equal sides
Perimeter of square
= sum of all sides of the square
= 4×length of a side of the square
Let, the length of each side be l unit, then
Perimeter of square = 4×l
p = 4×l
It is the formula for the perimeter of a square which is expressed as a relation between the perimeter and the length of the square. It is concise and easy to remember.
Here, p and l both are variables, l can take any value but value of p will depend on l.
Formula for Perimeter of a rectangle
A rectangle has two pairs of equal sides
Perimeter of rectangle
= sum of all sides of the rectangle
= length + breadth + length + breadth
= 2×(length + breadth)
Let, the length and breadth be l and b unit respectively, then
Perimeter of rectangle = 2×(l+b)
p = 2×(l+b)
It is the formula for the perimeter of a rectangle which is expressed as a relation between the perimeter and the length of the square. It is concise and easy to remember.
Here, p, l and b are variables, l & b can take any value independent of each other but value of p will depend on both l & b.
Some Rules from Arithmetic
Properties of NumbersProperties of numbers can be expressed concisely as a general rule by using variables -
Commutative Properties for addition and multiplication
Let a and b are two variables representing any two numbers, then
a+b = b+a
a×b = b×a
Distributivity of multiplication over addition
Let a, b and c are three variables representing any three numbers, then
a×(b+c) = a×b + a×c
a×(b−c) = a×b − a×c
Algebraic expressions
Arithmetic expression - Mathematical expression that contain numbers (constants) and operators (+, −, ×, ÷) are called arithmetic expressions. e.g., 2+5, 3×4-6 etc.Algebraic expression - Mathematical expression that contain both constants and variables together with operators (+, −, ×, ÷) are called algebraic expressions. e.g., 2×x+5, 3×4-6y etc.
In algebraic expressions we can use raised dot ⋅ or parenthesis ( ) in place of multiplication operator ×
When multiplying a number with a variable or a variable with another variable we can omit the multiplication operator.
2×x, 2(x), 2⋅ x and 2x are same expression.
A number expression like (3× 4) − 6 can be immediately evaluated as
(3 × 4) − 6
= 12 − 6
= 6
But an expression containing variable like (2x + 5), cannot be evaluated. Only if x is given some value, an expression like (2x + 5) can be evaluated. e.g., when x =3,
2x + 5 = 2⋅3 + 5 = 6 + 5 = 11
Forming Algebraic Expressions
Verbal phrases can be transformed into algebraic expressions by combining number and variables with operators.| (a) 7 added to p |
p+7
|
| (b) 7 subtracted from p |
p − 7
|
| (c) p multiplied by 7 |
p × 7
|
| (d) p divided by 7 |
p ÷ 7
|
| (e) 7 subtracted from – m |
−m − 7
|
| (f) – p multiplied by 5 |
−p × 5
|
| (g) – p divided by 5 |
−p ÷ 5
|
| (h) p multiplied by – 5 |
p(−5)
|
| (i) 11 added to 2m | 2m + 11 |
| (j) 11 subtracted from 2m | 2m − 11 |
| (k) 5 times y to which 3 is added | 5y + 3 |
| (l) 5 times y from which 3 is subtracted | 5y − 3 |
| (m) y is multiplied by – 8 | y(−8) |
| (n) y is multiplied by – 8 and then 5 is added to the result | −8y +5 |
| (o) y is multiplied by 5 and the result is subtracted from 16 | 16−5y |
| (p) y is multiplied by – 5 and the result is added to 16. | 16−5y |
Terms of an Expression | ↓Class 7 |
e.g., 2n has one term, 2n
5+ 3xy has two terms, 5 and 3xy
7x - 2y +4 has tree terms, 7x, −2y and 4
Factors of a term
A term is product of its factorse.g, factors of the term 2n are 2 and n
factors of the term 3xy are 3, x and y
factors of the term −2y are −2 and y
factors of the term 9x2 are 9, x and x
1 is not taken as separate factor.
Coefficients
The coefficient is the numerical factor in the term.e.g., coefficient of the term 2n is 2.
coefficient of the term −2y is −2.
coefficient of the term −5x2y is −5
When the coefficient of a term is +1, it is usually omitted.
e.g.,, 1x is written as x; 1 x2 is written as x2 and so on.
The coefficient (–1) is indicated only by the minus sign.
Thus (–1) x is written as – x; (–1)x2 is written as –x2 and so on.
Sometimes any factor or product of factors in a term is called the coefficient of the remaining part of the term.
e.g., coefficient of the y in the term −5x2y is −5x2
coefficient of the x2 in the term −5x2y is −5y.
Like & Unlike terms
Terms having same algebraic factors are called like terms. Thus, like terms have the same variables raised to the same power.e.g., xy , yx, -2xy, 3yx are like terms.
Terms having different algebraic factors are called unlike terms. Thus, unlike terms have different variables or same variables raised to different powers.
xy, yz, xy2, x2y are unlike terms.
Monomials and Polynomials
Polynomial
An algebraic expression containing one or more terms is called a polynomial in which the exponent (power) of the variables must be a whole number.
e.g., 2x, 5yz −2y, 2+x2−y+yx
2x−2−y is not a polynomial.
5√yz is not a polynomial.
e.g., 2x, 5yz −2y, 2+x2−y+yx
2x−2−y is not a polynomial.
5√yz is not a polynomial.
Monomial
A polynomial containing only one term is called monomial.e.g. 2x, 5yz, -7ab2c
Binomial
A polynomial containing two unlike term is called binomial.e.g. 2x3+3, 5y2z −2y, -7ab + c
Trinomial
A polynomial containing three unlike terms is called trinomial.e.g. 2+x2−y, x3+5y+z
Value of an Expression
The value of an expression depend on the value of variables it contain.We can find the value of an expression by putting the given value of variable in it.
Let's find the value of 2x−5 for x=7
Putting the value of x in 2x−5, we get
2x−5 = 2(7)−5 = 14−5 = 9
Addition & Subtraction of Algebraic Expressions
We can only add or subtract like terms, unlike terms cannot be added or subtracted with each other.While adding or subtracting the like terms, the variables (the common algebraic factors) are taken out and numerical factors or coefficient is then added or subtracted inside a parenthesis.
➢ 4x +7x +2y = x(4+7) + 2y = 11x + 2y
➢ 5y − 3y = y(5−3) = 2y
➢ Add 3x + 11 + 8z and 5x + 2z + 3
➥ (3x + 11 + 8z) + (5x + 2z + 3)
= 3x + 5x + 11 + 3 + 8z + 2z (rearranging the terms so that like terms can be grouped with each other)
= (3+5)x + (11+3) + (8+2)z
= 8x + 14 + 10z
➢ Subtract −m2 + 5mn from 4m2 − 3mn + 8
➥ (4m2 − 3mn + 8) − (−m2 + 5mn)
= 4m2 − 3mn + 8 + m2 − 5mn
= 4m2 + m2 − 3mn − 5mn + 8 (rearranging the terms so that like terms can be grouped with each other)
= (4+1)m2 −mn (3+5) + 8
= 5m2 −8mn + 8
➢ Simplify: (3y2 + 5y – 4) – (8y –y2 – 4)
➥ (3y2 + 5y – 4) – (8y –y2 – 4)
= 3y2 + 5y – 4 – 8y +y2 + 4
= 3y2 + y2 + 5y – 8y – 4+ 4
= (3+1)y2 + (5 – 8)y – (4– 4)
= 4y2 – 3y
While closing and opening the brackets, the signs of algebraic terms are handled in the same way as signs of numbers.