Introduction to Algebra
What is Algebra
Algebra is a branch of Mathematics that deals with symbols, letters and the arithmetic operations across these symbols.
It helps represent situations or problems in the form of Mathematical expressions.
It helps students learn how to think in a new way.It helps students organize their thinking, making it easier for them to craft reasonable responses when confronted with complicated or dynamic situations.
It helps represent situations or problems in the form of Mathematical expressions.
It helps students learn how to think in a new way.It helps students organize their thinking, making it easier for them to craft reasonable responses when confronted with complicated or dynamic situations.
Algebraic Expressions
In Algebra, Letters are used to stand for numbers. When letters are used to stand for different numbers, they are called Variables .
Usually, we use letters such as n, t, x, y, or z for Variable.
Usually, we use letters such as n, t, x, y, or z for Variable.
What is Algebraic Expressions
An Algebraic Expression is a collection of letters and symbols combined with atleast one of the arithmetic operations ( + , - , x , / )
for Example: 2x + y -3 and 5a3 b5 / (2x5 +5z) are Algebraic Expressions.
Note : that x (times) are not usually written in Algebraic Expressions, because, they might confuse learners to take them for the variable x.
To minimize that, we usually represent axb(which is a times b as), ab or (a)(b) or a.b
for Example: 2x + y -3 and 5a3 b5 / (2x5 +5z) are Algebraic Expressions.
Note : that x (times) are not usually written in Algebraic Expressions, because, they might confuse learners to take them for the variable x.
To minimize that, we usually represent axb(which is a times b as), ab or (a)(b) or a.b
Terms
In Algebra, Terms are separated by Plus (+) or Minus (-) signs.
For example: the terms of 3x-2y +5 are 3x, -2y and 5 . It means, there are three terms.
the terms of 5p4 +5q5r7 are 5p4 and 5q5r7 . It means, there are two terms.
Note: Once there is no Plus (+) or Minus (-) sign, then, It is one term.
< Remember > : The Positive (+) or Negative (-) sign can always be infront of the number. and anynumber without a sign, the default sign is + .
For Example: -3p6 has Negative (-) sign.
For Example: 8k2 has Positive (+) sign.
For Example, In : 2y3 - 11z8 , 2y3 has Positive (+) sign. and - 11z8 has Negative (-) sign.
A number on its own without any Variable is called a Constant Term
For Example : -5y2 + 11z3 -4 has -4 as a Constant Term because -4 has no Variable attached.
For Example :7p + 3-5y2 + 11z6 has +3 or 3 as a Constant Term because 3 has no Variable attached.
Note: A Term consist of a number, letters and Power or Exponent of the Letter.
For Example: 7y3 has number as 7, it has letter as y and Power or Exponent as 3
For Example: -4z2 has number as -4, it has letter as z and Power or Exponent as 2
< Remember > : It is always good to write a Term such that the number comes first, followed by the Letter.
Example: 3x2yz
Not x2y3z .
For example: the terms of 3x-2y +5 are 3x, -2y and 5 . It means, there are three terms.
the terms of 5p4 +5q5r7 are 5p4 and 5q5r7 . It means, there are two terms.
Note: Once there is no Plus (+) or Minus (-) sign, then, It is one term.
< Remember > : The Positive (+) or Negative (-) sign can always be infront of the number. and anynumber without a sign, the default sign is + .
For Example: -3p6 has Negative (-) sign.
For Example: 8k2 has Positive (+) sign.
For Example, In : 2y3 - 11z8 , 2y3 has Positive (+) sign. and - 11z8 has Negative (-) sign.
A number on its own without any Variable is called a Constant Term
For Example : -5y2 + 11z3 -4 has -4 as a Constant Term because -4 has no Variable attached.
For Example :7p + 3-5y2 + 11z6 has +3 or 3 as a Constant Term because 3 has no Variable attached.
Note: A Term consist of a number, letters and Power or Exponent of the Letter.
For Example: 7y3 has number as 7, it has letter as y and Power or Exponent as 3
For Example: -4z2 has number as -4, it has letter as z and Power or Exponent as 2
< Remember > : It is always good to write a Term such that the number comes first, followed by the Letter.
Example: 3x2yz
Not x2y3z .
Coefficient
The number placed infront of, and thus, multiplying a letter or group of letters is called The Coefficient
for Example: in 2x7 , 2 is the Coefficient of x Expressions.
for Example: in 3x2 -2y , 3 and -2 are the Coefficient of x and y respectively
for Example: in x3 -5y7 , 1 and -5 are the Coefficient of x and y respectively
This is because, any letter in Algebra without a Coefficient, the coefficient by default is 1.
This is why since the x has no coefficient physically expressed, the default Coefficient is 1.
for Example: in 3x3 - y7 , 3 and -1 are the Coefficient of x and y respectively
for Example: in 2x7 , 2 is the Coefficient of x Expressions.
for Example: in 3x2 -2y , 3 and -2 are the Coefficient of x and y respectively
for Example: in x3 -5y7 , 1 and -5 are the Coefficient of x and y respectively
This is because, any letter in Algebra without a Coefficient, the coefficient by default is 1.
This is why since the x has no coefficient physically expressed, the default Coefficient is 1.
for Example: in 3x3 - y7 , 3 and -1 are the Coefficient of x and y respectively
Power or Exponent
The Power or Exponent of a Term is the number raised on top of the Letter.
for Example: in 2x7 , 7 is the Power or Exponent of x
Note: 2x 7 is read as 2x raised to the 7th Power or 2x to the Exponent 8 .
for Example: in -3p2 -2y8 , 2 and 8 are the Power or Exponent of p and y respectively
for Example: in -11x3 - 5y , 3 and 1 are the Exponent of x and y respectively
This is because, any letter in Algebra without an Exponent or Power, the Exponent or Power by default is 1.
This is why since the y has no Exponent or Power physically expressed, the default Exponent is 1.
for Example: in 3p - y7 , 1 and 7 are the Exponent of p and y respectively
for Example: in 2x7 , 7 is the Power or Exponent of x
Note: 2x 7 is read as 2x raised to the 7th Power or 2x to the Exponent 8 .
for Example: in -3p2 -2y8 , 2 and 8 are the Power or Exponent of p and y respectively
for Example: in -11x3 - 5y , 3 and 1 are the Exponent of x and y respectively
This is because, any letter in Algebra without an Exponent or Power, the Exponent or Power by default is 1.
This is why since the y has no Exponent or Power physically expressed, the default Exponent is 1.
for Example: in 3p - y7 , 1 and 7 are the Exponent of p and y respectively
Liked Terms
Liked Terms are Terms that have the same letter (Variable) and raised to the same Power or Exponent
for Example: in 2x7 and -5x7 are Liked Terms .This is because, they have the same Variable x and the same Power or Exponent 7
for Example: in -2x8 and -5x10 are Not Liked Terms .This is because, The Power or Exponent are not the same.
for Example: in -2y7 and -5x7 are Not Liked Terms .This is because, The Variables are not the same.
Note: Before Terms becomes Liked Terms, They should meet two conditions :
1. They should have the same Variable (s)
2. They should have the same Power or Exponent (s)
If these conditions are not Met, they are not Liked Terms.
Example: 2x8y3 and -5x8y3 are Liked Terms .This is because, they have the same Variables x and y & the same Power or Exponent 8 and 3
for Example: in 2x7 and -5x7 are Liked Terms .This is because, they have the same Variable x and the same Power or Exponent 7
for Example: in -2x8 and -5x10 are Not Liked Terms .This is because, The Power or Exponent are not the same.
for Example: in -2y7 and -5x7 are Not Liked Terms .This is because, The Variables are not the same.
Note: Before Terms becomes Liked Terms, They should meet two conditions :
1. They should have the same Variable (s)
2. They should have the same Power or Exponent (s)
If these conditions are not Met, they are not Liked Terms.
Example: 2x8y3 and -5x8y3 are Liked Terms .This is because, they have the same Variables x and y & the same Power or Exponent 8 and 3
Exercise
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Self-Practice
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Answer To Exercise
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