Algebra

Algebra is one of the oldest branches of mathematics. It deals with theory, geometry and analysis. It also helps in the representation of the given situations in the form of mathematical expressions. It involves variables like x, y, and z. It follows mathematical operations such as addition, subtraction, multiplication and division.

Definition of Algebra

Algebra is the form of mathematics that deals with mathematical symbols and Operations related to these symbols. These mathematical symbols used do not have any particular value and are known as variables.

All the branches of mathematics such as trigonometry, calculus, coordinate geometry and more. An example of the algebraic expression is 2x + 80 = 156

Examples of Algebra

Let us take an example of Algebraic expression.

5x + 10 = 60

Here, x is the variable while 10 and 60 are the constants. In the above expression, the operation performed is of addition.

Branches of Algebra

On the basis of the use and complexity of the expressions, algebra can be classified into multiple branches. These branches of Algebra can be named as:

• Pre-algebra
• Elementary Algebra
• Abstract Algebra

Also Check: Classical Algebra

Pre-algebra

The basic algebra method of presenting a situation in the form of an algebraic expression with the help of variables is known as the pre-algebra method. It is helpful in transforming a real-life situation into mathematical equations.

Forming an algebraic expression out of the given mathematical problem is an example of Pre-algebra.

An example of a real-life mathematical expression is as follows.

Statement: The present age of Anjali is double the age of her daughter, Pihu. Ten years ago her age was four times the age of her daughter. Using the concept of algebra, find the present age of the daughter.

Solution: let the present age of the daughter be X years.

It is given that Anjali’s age is double that of her daughter. So, her age is 2X.

Now, for the situation 10 years ago, the age of her daughter was (X – 10) years and her age was (2X – 10). Also, her age then was 4 times that of her daughter.

Therefore, the equations formed are as follows.

2X – 10 = (4) (X – 10)

2X – 10 = (4X – 40)

2X – 4X = 10 – 40

-2X = -30

2X = 30

X = 30/2

X = 15

Therefore, the present age of her daughter is 15 years.

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Elementary Algebra

Elementary algebra focuses on solving algebraic expressions to find meaningful solutions. In this branch of mathematics, variables like x and y are typically represented in the form of equations.

Depending on the degree of these variables, equations can be classified into different types, such as linear equations, quadratic equations, and polynomials.

• Linear Equations: Linear equations are the simplest form, where the variable’s degree (the highest power of the variable) is 1. Some common forms of linear equations include:

ax + by = c

• Quadratic Equations: When the degree of the variable is 2, the equation is called a quadratic equation. The general form of a quadratic equation is:

ax2 + by = c

• Polynomials: For equations involving variables with degrees higher than 2, we refer to them as polynomial equations. The general form of a polynomial equation is:

axn + bxn+1 + cxn+2 + …… + k = 0

Here, n represents the degree of the polynomial, which determines the complexity of the equation.

As the degree increases, the methods for finding solutions often become more intricate. Elementary algebra provides the foundational skills needed to solve these equations and understand their relationships.

Abstract Algebra

Abstract algebra is a branch of mathematics that focuses on the study of concepts like groups, rings, and vectors, rather than dealing with basic arithmetic or elementary algebraic equations. It explores mathematical structures that go beyond simple number systems, providing a deeper understanding of how these structures interact.

Applications of Abstract Algebra

Abstract algebra has numerous practical applications:

• Computer Science: Abstract algebra is used in cryptography, coding theory, and algorithms, where the properties of algebraic structures help in designing secure and efficient systems.
• Physics: Group theory, in particular, is used to study symmetries and conservation laws in physics.
• Astronomy: Abstract algebra aids in the modelling of celestial mechanics and the understanding of symmetries in astronomical objects.
• Engineering: Vector spaces and other algebraic structures are crucial in the analysis and solution of engineering problems, particularly in control theory and signal processing.

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Algebra Formulas

Algebraic identities are fundamental equations that hold true for all values of the variables involved. These identities are useful for simplifying expressions, solving equations, and performing various algebraic manipulations. The major algebraic identities in algebra are discussed below:

• (a+b)² = a²+2ab+b²
• (a-b)² = a²- 2ab+b²
• (a+b)(a-b) = a²-b²
• (x+a)(x+b) = x²+x(a+b)+ab
• (a+b)³ = a³+b³+3ab(a+b)
• (a-b)³ = a³-b³-3ab(a-b)
• (a+b+c)² = a²+b²+c²+2(ab+bc+ca)
• a³+b³+c³-3abc = (a+b+c)(a²+b²+c²-ab-bc-ca)

Algebra Formulas PDF Download

The formulas used in algebra form the basis of many major concepts. Therefore, one must keep all these formulas handy. Use the link below to download the Algebra formulas and identities in PDF Format.

Algebra formulas and identities PDF

Algebra Operations and Examples

In algebra, the fundamental operations are addition, subtraction, multiplication, and division. Below discussed are the operations, notations and examples of these operations.

1. Addition

Operation: The role of this algebraic operation is to combine two or more algebraic expressions by adding them together.

Notation: Expressions are joined with a plus (+) sign.

Example: 2x + 5x = 7x

2. Subtraction

Operation: The role of this algebraic operation is to find the difference between two or more algebraic expressions.

Notation: Expressions are separated by a minus (-) sign.

Example: 10x – 8x = 2x

3. Multiplication

Operation: The role of this algebraic operation is to multiply two or more algebraic expressions together.

Notation: Expressions are connected with a multiplication (×) sign or by placing them next to each other.

Example: 3x 5y = 15xy

4. Division

Operation: The role of this algebraic operation is to Divide one algebraic expression by another.

Notation: Expressions are separated by a division (/) sign or represented as a fraction.

Example: 50x5x = 10

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Algebra Solved Examples

1. Solve the following: 2X – 10 = (4) (X – 10)

Solution: 2X – 10 = (4) (X – 10)

2X – 10 = (4X – 40)

2X – 4X = 10 – 40

-2X = -30

2X = 30

X = 30/2

X = 15

2. Solve the following: 5y – 15 = (6) (y – 5)

Solution: 5y – 15 = (6) (y – 5)

5y – 15 = (6y – 30)

5y – 6y= (15 – 30)

-y = -15

y = 15

Algebra Practice Questions

1. Solve the following: 4x – 2 = (8) (x – 1)
2. Solve the following: 7y – 16 = (4) (y – 8)
3. Solve the following: 2x – 10 = (8) (x – 5)
4. Solve the following: 27z – 3 = (9) (z – 1)

FAQs on Algebra

What is called algebra?

Algebra is a branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of statements within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations. Elementary algebra, also called school algebra, college algebra, and classical algebra, is the oldest and most basic form of algebra. It relies on the same operations as arithmetic while allowing variables in addition to regular numbers. Abstract algebra, also called modern algebra, studies different types of algebraic structures.

How to introduce algebra?

To introduce algebra, it's helpful to start with basic definitions and concepts. Variables are symbols, usually letters like x or y, that represent quantities without fixed values. Coefficients are the number parts of terms with variables. Constants are terms that contain only numbers and never change. Equations describe relationships between variables. Evaluating an algebraic expression means substituting specific values for its variables.

What is algebra basics?

The basics of algebra include manipulating algebraic expressions, often with the purpose of solving equations. An equation is an assertion that two algebraic expressions are equal. Solving an equation means finding the values of the variables that make the equation true. Linear equations and quadratic equations are two common types. Algebra also involves the study of systems of linear equations, which can be solved using methods like Gaussian elimination.

What is the golden rule of algebra?

The golden rule of algebra is to maintain equality when solving equations. This means performing the same operations to both sides of an equation to isolate the variable. For example, to solve 4x - 7 = 5 for x, you would add 7 to both sides to get 4x = 12, then divide both sides by 4 to get x = 3.

What is ABC algebra?

ABC algebra is a mnemonic device to help remember the order of operations in algebra: A for addition, B for brackets, and C for coefficients. This refers to the order in which operations should be performed: first, evaluate expressions inside brackets; second, evaluate coefficients and exponents; and third, perform addition and subtraction from left to right.