Basic algebra

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BASIC ALGEBRA

What is Algebra? – is a way of expressing mathematical ideas
that use variables, constants, numerical coefficients, and
mathematical operations.
ALGEBRA
© 2019 eTutorWorld. All rights reserved.
What is a Variable?
It is a letter that represents an unknown value or a changing value.
all letters like x, y, a, z, m
What is a Constant?
A quantity which is not changed.
all numbers like 3, -6, 14, 23, 100

ALGEBRA
What is an Algebraic Expression?
A mathematical statement that contains terms (numbers and
variables) and at least one operation ( +, -, ÷, ×)
Examples of Algebraic Expression: x2, 2y2, 3x2 – 5, 4xy + 7
What is a Term?
It is made up of a number and a variable put together by multiplication
and separated by addition or subtraction. There can be two or more
terms in an expression.
5y has one term, 5x + 3y has two terms, 3x2 - 7ab + 2e√π
has three terms and so on

ALGEBRA
What is a Coefficient?
Coefficient of x2 in 3x2 is 3
Coefficient of y9 in 0.25y9 is 0.9
Coefficient of m in -25m is -25
Coefficient of x2y in 7x2y is 7
It is the number part of the terms that multiply a variable or powers
of a variable . In simple terms, it is the number before a variable in
an expression

ALGEBRA
What is a Monomial?
An algebraic expression which consists of one non-zero term only is
called a monomial.
Examples of a monomial: x, 10ab2, 5m2n, -16b3
What is a Binomial?
An algebraic expression which consists of two non-zero terms is
called a binomial.
Examples of a binomial: m + n, a2 + 3b, 5x3 – 9y2, 5m2n2 + 1/7
What is a Trinomial?
An algebraic expression of three non-zero terms only is called a trinomial.
Examples of a trinomial: x + y + z, 2a2 + 5a + 7, 5abc – 7ab + 9ac

ALGEBRA
What is a Polynomial?
An algebraic expression which consists of more than two terms is
called a polynomial.
Examples of polynomial: 2a + 5b, 3xy + 5x + 1, 3y4 + 2y3 + 7y2 - 9y – 17
What should a polynomial not contain?
Polynomials should not contain any term whose degree is a negative
number and or a radical.
How can we identify a Polynomial?
(i) 3xy-2 is not, because the exponent is "-2" (exponents can only
0,1,2,...)
(ii) 2/(x+2) is not, because dividing by a variable is not allowed
(iii) 1/x is not either
(iv) √x is not, because the variable has a radical sign or exponent is
"½"

ALGEBRA
What is difference between a polynomial and an algebraic expression?

ALGEBRA
How can we identify the Degree of an Algebraic Expression
or Polynomial?
The degree of the expression 2x2 - 3x5 + 5x6 has the highest power
____ and hence the degree is ____
The degree of the expression 2x2 - 3x + 5 has the highest power
____ and hence the degree is ____
The degree of the expression 2xy - 3a - 5b + 7 has the highest
power ____ and hence the degree is ____
2x2 - 3x5 + 5x6 : 6
2x2 - 3x + 5 : 2
2xy - 3a - 5b + 7 : 2
The degree of the polynomial is the greatest of the exponents
(powers) of its various terms. If there are two variables together,
we add the powers and consider the degree.

ALGEBRA
What are Like terms?
x2 and -5x2, 3x4 and -9x4, 6xy and 13xy
Two or more terms are called like terms if they have same variable
with same power
What are Unlike terms?
Two or more terms are called unlike terms if they have one of the
following:
(i) same variable with different power : x2 and 3x
(ii) different variable with same power : y4 and x4
(iii) different variable with different power : a3 and b2

ALGEBRA
What are Factors of an Algebraic Expression?
Example:
The factors of 4m3 is
The factors of 27a5 is
The factors of 10p2 is
The factors of 60m2n3y is 2 x 2 x 3 x 5 x m x m x n x n x n x y.
The letter and/or numbers multiplied together are factors of the term.
Standard form of expression
Example: 3x2 − 7 + 4x3 + x6 is written in standard form as
x6 + 4x3 + 3x2 − 7
A way of writing a polynomial with terms having highest degree first,
then next degree and so on and constant in the last.

ALGEBRA
Evaluating Algebraic expressions
(i) The value of the expression x2 - 6xy + 9y2 when x = 2 and
y = -1 is (2)2 – 6(2)(-1) + 9(-1)2 = 4 + 12 + 9 = 25
(ii) The value of the expression (6 + h2 - j) ÷ 2 when h = 6 and
j = 4 is [6 + (6)2 – 4) ÷ 2 = 38 ÷ 2 = 19
To evaluate an algebraic expression means to replace(substitute) the
variables in the expression with numeric values that are assigned to
them and perform operations in the expression.

ALGEBRA
How can you convert a word phrase into an algebraic
expression?
Phrases used:
Addition : added to, sum, total, more than, increased by
Subtraction : subtracted from, take away, less than, decreased by,
difference
Multiplication : multiplied by, product of, times
Division : Divided by, quotient of
Examples:
(i) The equivalent algebraic expression for “6 less than 3 times a
number” is
(ii) The equivalent word phrase for “4p/5 + 3h/5 + 8” is

ALGEBRA
What is a Simple/Linear equation?
Simple equation is an equation in which two algebraic expressions are
made equal to each other. The degree of the expression is always 1
Examples of Linear Equation:
Solve for x :
2x – 19 = 5

© 2018 eTutorWorld. All rights reserved.Preparation for Math Olympiad
CLASS VI - ALGEBRA
Practice questions
1) Coefficient of x in √25 x is
a) 5 b) 25 c) 0.125 d) 0.25
2) The equation representing the statement “The quotient of
fifty and
five more than a number is ten” is
a) 50 + 5/n = 10 b) 50/(n + 5) = 10
c) n = 50/5 + 10 d) 50 + n/5 = 10
Mathematical reasoning

CLASS VI - ALGEBRA
3) The value of the expression 7c + 2(-5f – 3) when f = -4 and
c = 6
a) 19 b) 1 c) 34 d) 76
4) The value of m in the equation 16(m + 7) = 144 is
a) 2 b) 1 c) 0 d) 3

5) When Alex multiplies a certain number by 15 and
subtracts 4 from the product, he gets 326. Find the number.
a) 14 b) 15 c) 22 d) 60
6) 24a – [3(3a – 3) – 4(5a)] - 9 is equal to ?
a) 0 b) 1 c) 20a d) 35a
CLASS VI - ALGEBRA
Logical reasoning

CLASS VI - ALGEBRA
7) I had Rs. 100. I gave Rs y to Abhishek, Rs y/2 to Praveen
and I am left with Rs. 55. How much did I give to Praveen?
a) Rs. 15 b) Rs. 35 c) Rs. 45 d) Rs. 55
8) The measure of second side of a triangle is twice the
first side and the third side of the triangle is thrice the
second side. The perimeter of the triangle is 72 cm. What is
the measure of the third side?
a) 20 b) 16 c) 48 d) 8
Everyday Mathematics
Logical reasoning

CLASS VI - ALGEBRA
9) Ages of two persons are in the ratio 5 : 2. If the sum of
their ages is 49, then their ages are respectively
a) 25 and 24 b) 35 and 14 c) 30 and 29 d) 31 and 18
10) Sum of three consecutive numbers is 126. Find the greatest
number
a) 41 b) 44 c) 42 d) 43
Everyday Mathematics
Logical reasoning

CLASS VI - ALGEBRA
11) State which one of the following given in the brackets is a
solution to the given equation
a) 2n + 12 = 19(n = 1) b) 4p – 3 = 13(p = -4)
c) 7n - 5 = -19(n = -2) d) 4p – 3 = 13(p = 0)
a) 45 b) 30 c) 20 d) 35
12) Find the measurement of the smallest angle
Logical reasoning

CLASS VI - ALGEBRA
13) Hardeep thinks of a number. If she adds 19 to it and divides
the sum by 5, she will get 8. What is the number?
a) 21 b) 33 c) 38 d) 42
14) Mr. Surendar sells mangoes in two types of boxes, one
small and one large. A large box contains as many as 8 small
boxes plus 4 loose mangoes. If the number of mangoes in a
large box is 100, how many mangoes are there in small box?
a) 6 b) 8 c) 10 d) 12
Achiever’s section
Logical reasoning

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Preparation for Math Olympiad

Explanation on examples (for tutors only)
Variables : all letters like x, y, a, z, m
Constants : all numbers like 3, -6, 14, 23, 100
Algebraic expression : x2, 2y2, 3x2 – 5, 4xy + 7
Term : 5y has one term, 5x + 3y has two terms, 3x2 - 7ab + 2e√π
has three terms and so on
Coefficient :
Coefficient of x2 in 3x2 is 3
Coefficient of y9 in 0.25y9 is 0.9
Coefficient of m in -25m is -25
Coefficient of x2y in 7x2y is 7
Monomials : x, 10ab2, 5m2n, -16b3
CLASS VI - ALGEBRA

Binomials : m + n, a2 + 3b, 5x3 – 9y2, 5m2n2 + 1/7
Trinomials : x + y + z, 2a2 + 5a + 7, 5abc – 7ab + 9ac
Polynomial : 2a + 5b, 3xy + 5x + 1, 3y4 + 2y3 + 7y2 - 9y – 17
Polynomial should not contain the following :
Polynomials should not contain negative power and radicals
(i) 3xy-2 is not, because the exponent is "-2" (exponents can only 0,1,2,...)
(ii) 2/(x+2) is not, because dividing by a variable is not allowed
(iii) 1/x is not either
(iv) √x is not, because the variable has a radical sign or exponent is "½"
Degree of the expression for the following expressions :
2x2 - 3x5 + 5x6 : 6
2x2 - 3x + 5 : 2
2xy - 3a - 5b + 7 : 2
CLASS VI - ALGEBRA

Like terms : x2 and -5x2, 3x4 and -9x4, 6xy and 13xy
Unlike terms :
(i) same variable with different power : x2 and 3x
(ii) different variable with same power : y4 and x4
(iii) different variable with different power : a3 and b2
Factors of the term :
The factors of 60m2n3y is 2 x 2 x 3 x 5 x m x m x n x n x n x y
Standard form of expression 3x2 − 7 + 4x3 + x6 is x6 + 4x3 + 3x2 − 7
Evaluating algebraic expressions :
The value of the expression x2 - 6xy + 9y2 when x = 2 and y = -1
is (2)2 – 6(2)(-1) + 9(-1)2 = 4 + 12 + 9 = 25
The value of the expression (6 + h2 - j) ÷ 2 when h = 6 and j = 4
is [6 + (6)2 – 4) ÷ 2 = 38 ÷ 2 = 19
ALGEBRA

Converting word phrase into algebraic expression:
(i) The equivalent algebraic expression for “6 less than 3 times a number”
3x - 6
(ii) The equivalent word phrase for “4p/5 + 3h/5 + 8” is
The sum of four fifth of p, three fifth of h and 8
Simple Linear equation :
2x – 19 = 5
Add 19 to both sides, we get
2x – 19 + 19 = 5 + 19
2x = 24
Divide by 2 on both sides, we get
x = 12
CLASS VI - ALGEBRA

1) a
Explanation : The square root of 25 is 5
2) b
3) d
Explanation : 7c + 2(-5f – 3) = 7(6) + 2[-5(-4) - 3] = 42 + 2(17) = 76
4) a
Explanation : 16(m + 7) = 144
=> (m + 7) = 144/16 = 9
=> m = 2
5) c
Explanation : Let the number be m.
So 15m - 4 = 326
=> 15m = 330
=> m = 22
ANSWER KEY
CLASS VI - ALGEBRA

6) d
Explanation : 24a – [3(3a – 3) – 4(5a)] - 9 = 24a - (9a - 9 - 20a) - 9 =
24a - (-11a - 9) - 9 = 24a + 11a + 9 - 9 = 35a
7) a
Explanation : 100 - y - y/2 = 55
=> 200 - 3y = 110
=> 3y = 90
=> y = 90/3 = 30
=> Praveen got y/2 = 30/2 = Rs. 15
8) c
Explanation : Let the measure of the first side of the triangle be m
So m + 2m + 6m = 72
=> 9m = 72 and so m = 72/9 = 8
Measure of the third side = 6m = 6(8) = 48 cm.
ANSWER KEY
CLASS VI - ALGEBRA

9) b
Explanation : According to the question, 5x + 2x = 49
=> 7x = 49 and so x = 49/7 = 7
So age of first person is 5(7) = 35 and age of second person is 2(7) =
14.
10) d
Explanation : Let first number be x
So x + (x + 1) + (x + 2) = 126
=> 3x + 3 = 126
=>3x = 123
=> x = 123/3 = 41
third number is x + 2 = 41 + 2 = 43
11) c
Explanation : 7(-2) - 5 = -19
=> -14 - 5 = -19.
=> -19 = -19. Balanced
ANSWER KEY
CLASS VI - ALGEBRA

12) b
Explanation : We know 2x + 3x + 7x = 180
=> 12x = 180
=> x = 180/12 = 15
Smallest angle = 2x = 2(15) = 300
13) a
Explanation : Let hardeep think m
So (m + 19)/5 = 8
=> m + 19 = 8 x 5 = 40
=> m = 40 - 19 = 21
14) d
Explanation : Let the number of mangoes in small box be y
So large box contains 8y + 4
According to the question, 8y + 4 = 100
=> 8y = 100 - 4 = 96
=> y = 96/8 = 12
ANSWER KEY
CLASS VI - ALGEBRA

Basic algebra

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