Algebra for Class 9

Algebra is known as a discipline of mathematics that deals with symbols and the operations that may be performed on them. These symbols which are used are referred to as variables because they do not have any set values. We use the encounter specific values that change in our real-life question. These are commonly represented in algebra by symbols such as x, y, z, p, or q, and these symbols are referred to as variables. Furthermore, these symbols are subjected to a variety of arithmetic operations, including addition, subtraction, multiplication, and division, with the goal of determining the values.

•  Branches of Algebra:

The employment of more than one algebraic expression reduces the complexity of algebra. Algebra can be divided into a few branches based on how expressions are used and how complex they are. These branches are listed below:

• Pre-algebra
• Elementary Algebra
•  Abstract Algebra
• Universal Algebra

• Pre-algebra

The fundamental methods for presenting unknown values as variables aid in the creation of mathematical statements. It aids in the transformation of real-world issues into algebraic expressions in mathematics. Pre-algebra involves formulating a mathematical expression for the given problem statement.

• Elementary Algebra

Solving algebraic expressions for a feasible answer is the focus of elementary algebra. Simple variables like x and y are expressed as equations in elementary algebra. The equations are classified as linear equations, quadratic equations, or polynomials according to the degree of the variable. a x + b = c, ax + by + c = 0, ax + by + cz + d = 0 are examples of linear equations. The degree of the variables in elementary algebra leads to quadratic equations and polynomials. A quadratic equation is represented as ax2 + bx + c = 0, while a polynomial equation is represented as axn + bxn-1+ cxn-2+….k = 0.

• Abstract Algebra

Rather than using simple mathematical number systems, abstract algebra employs abstract ideas such as groups, rings, and vectors. The addition and multiplication properties are written together to form rings, which is a simple level of abstraction. Abstract algebra includes two essential concepts: group theory and ring theory. Abstract algebra, which employs vector spaces to express quantities, has many applications in computer science, physics, and astronomy.

• Universal Algebra

Universal algebra encompasses all other mathematical forms involving trigonometry, calculus, and coordinate geometry involving algebraic expressions. Throughout these topics, universal algebra focuses on mathematical expressions rather than algebraic models. Universal algebra can be regarded as a subset of all other areas of algebra. Any real-world problem can be categorised into one of the fields of mathematics and solved with abstract algebra.

• Algebraic Expressions

In algebra, an algebraic expression is created using integer constants, variables, and the addition(+), subtraction(-), multiplication(x), and division(/) arithmetic operations. 5x + 6 is an example of an algebraic expression. 5 and 6 are constants, but x is a variable. Furthermore, the variables can be simple alphabetic variables such as x, y, and z, or complex variables such as x2, x3, xn, xy,x2y, and so on. Polynomials are another name for algebraic expressions. Variables (also known as indeterminates), coefficients, and non-negative integer exponents of variables make up a polynomial. For instance, 5×3 + 4×2 + 7x + 2 = 0.

A maths statement having a ‘equal to’ sign between 2 algebraic expressions with equal values is known as an equation. The following are the various types of equations that we can use algebra to solve, dependent on the degree of the variable:

Linear Equations: Linear equations are used to represent the relationship between variables such as x, y, and z, and are stated in one-degree exponents. We utilise algebra to solve these linear equations, starting with the fundamentals like adding and subtracting algebraic expressions.

Quadratic Equations: In standard form, a quadratic equation is represented as ax2 + bx + c = 0, where a, b, and c are constants and x is the variable. Solutions of the equation are the values of x that fulfil the equation, therefore a quadratic equation can only have two solutions.

Cubic Equations: Cubic equations are algebraic equations in which the variables have a power of three. ax3 + bx2 + cx + d = 0 is a generalised form of a cubic equation. In calculus and three-dimensional geometry, a cubic equation has several uses (3D Geometry).

• Algebraic Formulas 

An algebraic identity is a formula that is always true, regardless of the variables’ values. For all values of the variables, identity indicates that the left-hand side of the equation is identical to the right-hand side. These formulae involve squares and cubes of algebraic expressions and aid in the speedy solution of algebraic expressions. The following are some of the most commonly used algebraic formulas.

• (a + b)2 = a2 + 2ab + b2
• (a – b)2 = a2 – 2ab + b2
• (a + b)(a – b) = a2 – b2
• (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
• (a + b)3 = a3 + 3a2b + 3ab2 + b3
• (a – b)3 = a3 – 3a2b + 3ab2 – b3

• Algebraic operations

Addition, subtraction, multiplication, and division are the four basic processes covered in algebra.

In algebra, the addition operation is performed by separating two or more equations with a plus (+) symbol.

In algebra, the subtraction operation is performed by separating two or more expressions with a negative (-) sign.

In algebra, a multiplication (x) sign separates two or more equations for the multiplication operation.

In algebra, the division operation is performed by separating two or more expressions with a “/” symbol.

• Properties of algebra:
• Commutative Property of Addition: a + b = b + a
• Commutative Property of Multiplication a × b = b × a
• Associative Property of Addition: a + (b + c) = (a + b) + c
• Associative Property of Multiplication: a × (b × c) = (a × b) × c
• Distributive Property: a × (b + c) = (a × b) + (a × c), or, a × (b – c) = (a × b) – (a × c)
• Reciprocal: Reciprocal of a = 1/a
• Additive identity a + 0 = 0 + a = a
• Multiplicative identity: a × 1 = 1 × a = a
• Additive inverse: a + (-a) = 0

Conclusion: 

Algebra is a discipline of mathematics that uses mathematical expressions to represent issues. To make a meaningful maths statement, it uses variables like x, y, and z, as well as mathematical operations like addition, subtraction, multiplication, and division. Elementary algebra, abstract algebra, linear algebra, Boolean algebra, and universal algebra are different types of algebra. These are named after the problems that we can solve with the help of algebra.