Class 7 – Algebra

Take practice tests in Algebra

laptop_img Online Tests

Topic Sub Topic Online Practice Test
Algebra
  • Review of algebra
  • Algebraic expressions and polynomials
  • Addition and subtraction of polynomials
  • Multiplication and division of polynomials
Take Test See More Questions
Algebra
  • Simplification of brackets
  • Special products and identities
Take Test See More Questions
Algebra
  • Factorization
  • Framing a formula
Take Test See More Questions
Algebra
  • Linear equations in one variable
  • Linear inequations in one variable
Take Test See More Questions

file_img Study Material

Introduction: Algebra, a branch of Mathematics was believed to be started more than 3500 years ago, around 1550 BC, when people in Egypt started using symbols to denote unknown numbers.

Around 300 BC, use of letters to denote unknown and forming expressions from them was quite common in India. Many great Indian mathematicians, Aryabhatt (born 476 AD), Brahmagupta (born 598 AD), Mahavira (who lived around 850 AD) and Bhaskara II (born 1114 AD) and others, contributed a lot to the study of algebra.

The word ‘algebra’ is derived from the title of the book, Hidab al-jabar wal-muqubala written about 825 AD by an Arab mathematician, Mohammed Ibn-musa Al Khowarizmi of Baghdad.

Numerical expressions:
A combination of numbers connected by one or more of the symbols +, –, × , ÷ is called a numerical expression.

Example:  is an example of numerical expression.

Algebra: Algebra treats quantities as in arithmetic, but these algebraic quantities are denoted by symbols which may have any value we choose to assign to them.

http://images.clipartpanda.com/bloc-clipart-note-md.png

i)      The symbols used in Algebra are generally letters of our own alphabet such as a, x, y, etc. except ‘O’ as it resembles zero.

ii)     There is no restriction as to the numerical values a symbol may represent.

 

Constant and Variables: A basic characteristic of algebra is the use of symbols (usually letters) to represent numbers.

Variable: A letter or symbol that represents any member of a collection of two or more numbers is called a Variable. Generally the letters x, y, z … are used to represent variables.

Constant: A letter or symbol that represents a specified number, known or unknown is called a Constant.

Example: Suppose the streets on your way from home to school have speed limits of 40 kmph, 55 kmph, 60 kmph. In algebra we can let the letter “x” represent the speed as you travel from home to school. The letter x can assume any of the values 40, 55, 60 and its maximum value depends on the street you choose to walk. Hence x is treated as a variable.

Example: Suppose that in writing a term paper for a geography class we need to specify the height of Mount Everest. If we do not know the height of the mountain, we can represent it (at least temporarily) on the paper with the letter h. Later we can find it to be 8,848 meters. The letter h can assume only one value 8,848 and no others. The value h is thus treated as a constant.

Operations on literals (or) variables and constants:

Addition of literals and numbers:

i) 10 added to a is written as a + 10;

ii) added to x is written as x + y.

The properties of addition of numbers also hold good in algebra:

Thus, for any literals x, y, z we have:

i)                          (Commutative law)

ii)                 (Associative law)

iii)                         (Additive identity)

iv)                 (Additive inverse)

Subtraction of literals and numbers:

i) p is decreased by 6 is written as p – 6.

ii) Subtract y from 9 is written as 9 – y.

iii) x is decreased by y is written as x – y.

Multiplication of literals and numbers:

i)     5 times b is written as 5 × b denoted by 5 × b or 5b not b5.

ii)   The product of x and y is x × y, written as x.y or xy.

iii) 1 × x is written as x and not 1x. similarly (-1) × x is written as –x.

The properties of multiplication of numbers also hold good in algebra:

Thus, for any literals x, y, z we have:

i)                                 (Commutative law)

ii)                     (Associative law)

iii)                   (Multiplicative  identity )

iv)               (Distributive Property)

v)                   (Property of zero)

vi)                (Multiplicative inverse)

Division of literals and numbers:

i)     8 divided by b is written as

ii)   g divided by 4 is written as

iii) x divided by y is written as .

Practical use of Expressions:

Example:

Situation (described in ordinary languages)

Variable

Statements using expressions

How old will Devi be 5 years from now?

Let y be Devi’s present age in years

Five years from now Devi will be  years old

How old was Ram 4 years ago?

Let y be Ram’s present age in years.

Four years ago, Ram was

 years old

Price of wheat per kg is ` 5 less than price of rice per kg.

Let price of rice per kg be ` p.

Price of wheat per kg is ` (p – 5)

Price of oil per litre is 5 times the price of ghee per kg.

Let price of ghee per kg be ` p.

Price of oil per litre is ` 5p.

The speed of a bus is 10 kmph more than the speed of a truck going on the same road

Let the speed of the truck be y kmph

The speed of the bus is  kmph

 

Bases and Exponents: An exponent is a positive or negative number or 0 placed above and to the right of a quantity. It expresses the power to which the quantity is to be raised or lowered.

Example: In 34, 3 is the base and 4 is the exponent. It shows that 3 is to be used as a factor four times: (multiplied itself four times). 34 is read as “three is raised to the power 4 ” (or simply three to the power four)

Similarly .

In general, if x is any real number and n is a natural number then

        

http://images.clipartpanda.com/bloc-clipart-note-md.pngIf x is any non zero number (i.e., x ≠ 0) then x1 = x and x0 = 1

 

Example: 51 = 5; 50 = 1 and 61 = 6; 60 = 1.

When the expression appears in a continuous row, we have to know which sign to be operated first. This order of operation is explained using PEMDAS rule.
Where P – ‘Parenthesis’, E – ‘Exponents’, M – ‘Multiplication’, D – ‘Division’, A – ‘Addition, S – ‘Subtraction’.

Order of Operations:

i)     Perform all the operations inside grouping symbols beginning from the innermost set.

ii)   Perform all the exponential operations as you come to them, moving from left to right.

iii) Perform all multiplication and division as you come to them moving from left to right.

iv)  Perform all addition and subtraction as you come to them, moving from left to right.

Example: Simplify the following
 

 

Laws of exponents:

1.   The product rule of exponents: To understand the rule for multiplying two exponential quantities having same base and different natural exponents let us see the following examples
Example:


Thus, the product rule of exponents is
If x is a real number and m, n are natural numbers,
Rule1:

To multiply two exponential quantities having the same base, add the exponents.
Example:

http://images.clipartpanda.com/bloc-clipart-note-md.png The product rule does not apply as the bases are different.

 

2.   The Quotient rule of exponents: To understand the rule for the quotient of two exponential quantities having same base and different natural exponents let us see the following examples
Example: (Notice that 7 – 5 = 2)
Thus, the Quotient rule of exponents is
If x is a real number and m, n are natural numbers.
Rule 2: . To divide two exponential quantities having the same non-zero base, subtract the exponent of the denominator from the exponent of the numerator.
Example:

When we make the subtraction, n – m, in the division, there are three possibilities for the value of exponents:

1. The exponent of the numerator is greater than the exponent of the denominator, that is, m > n. Thus, the exponent, m – n is a natural number.

2. The exponents are the same, that is n = m. Thus, the exponent m – n is zero, a whole number.

3. The exponent of the denominator is greater than the numerator that is
n > m. Thus, the exponent of the denominator n-m is an integer.

Rule 3:

Example:

               

Rule 4: , that is

Example:

               

3.   Reciprocal: The reciprocal of non-zero integer x is denoted by x-1 and defined as . For a fractional number  (where) we have.
The reciprocal of is given by .

http://www.aids.harvard.edu/img/news/spotlight/newsletter_layout/logo.gif For three different numbers x, y and z, we have

(i)     (ii)       (iii)         (iv)

(v)         (vi) If  then m = n
Where m and n are positive integers.

 

Example: Simplify

      

Solutions:

            [Using ]

                       [Using   ]

        

        

        

        

        

Example: Solving exponential equations.

Solution:

        

       

Here bases are equal, powers can be equated. So,