Careers360 Logo
Algebraic Identities for Class 9 With Proofs and Examples

Algebraic Identities for Class 9 With Proofs and Examples

Edited By Team Careers360 | Updated on Jan 28, 2024 07:22 PM IST

In algebra, which is a branch of mathematics, quantities and numbers are represented by generic symbols and letters in equations and formulas. Algebra is broadly divided into two branches: Elementary algebra and Modern algebra, also known as Abstract algebra. The fundamentals of higher-level algebraic studies are algebraic formulas. To do this, one must have sound knowledge of how to comprehend and resolve algebraic expressions. Numerous types of mathematical representations are used in algebra, including real numbers, complex numbers, vectors, matrices, and more.

What Are Algebraic Identities?

Algebraic identities are algebraic equations that are true irrespective of the value of each variable. Additionally, they are useful in the factorization of polynomials. An equation is said to be an algebraic identity if, for all values of the variables, the value on the left side of the equation equals the value on the right side. Calculating algebraic expressions and solving various polynomials both involve the usage of algebraic identities. Equations incorporating numbers, variables, and mathematical operators like addition, subtraction, multiplication, and division are known as algebraic identities and expressions..

Properties Of Algebraic Identities

  • Algebraic identities are equations in algebra that persist irrespective of the value of each of its variables.

  • The factorization of polynomials involves the use of algebraic identities.

  • On both sides of the algebraic identity, they possess variables and constants.

  • The left and right sides of an equation are equal in an algebraic identity.

  • An algebraic Identity is a universal equality that applies to all values.

  • If the values of the variables are altered, the value of an algebraic identity will also change.

  • Either a substitution method or basic geometrical models can be used to confirm these algebraic identities.

How To Verify Algebraic Identities?

There are two simple approaches for verifying algebraic identities.

The substitution approach is used to validate the algebraic identities. Apply the arithmetic operation using the substituted values in place of the variables in this technique. The activity technique is another means of confirming the algebraic identity.

Substitution Method

In general, substitution refers to replacing variables or characters with numbers or values.

The substitution method involves altering the values for the variables to carry out an arithmetic operation.

Any value of the variable will hold true for both the left and right sides of the equation if you have expanded or correctly solved an example utilizing algebraic identities.

Activity Method

Using various x and y values, the algebraic identity is geometrically confirmed using this method.

By cutting and pasting pieces of paper, the identities are validated using the activity technique.

You need a fundamental understanding of geometry in order to use this method of identity verification.

Algebraic Identities For Two Variables

Here is the list of algebraic identities involving two variables,

(x+y)^{2}=x^{2}+2xy+y^{2}

1706449745318

(x-y)^{2}=x^{2}-2xy+y^{2}

1706449745473

x^{2}-y^{2}=(x+y)(x-y)

1706449745085

(x+a)(x+b)=x^{2}+(a+b)x+ab

1706449744154

(x+y)^{3}=x^{3}+y^{3}+3xy(x+y)

1706449744606

(x-y)^{3}=x^{3}-y^{3}-3xy(x-y)

1706449744425

Algebraic Identities For Three Variables

Here is the list of algebraic identities involving two variables,

(x+y+z)^{2}=x^{2}+y^{2}+z^{2}+2xy+2yz+2zx

1706449744087

x^{3}+y^{3}+z^{3}-3xyz=(x+y+z)(x^{2}+y^{2}+z^{2}-xy-yz-zx)

1706449743948


Proof Of Algebraic Identities

  • (x+y)^{2}=x^{2}+2xy+y^{2}

JEE Main Highest Scoring Chapters & Topics
Just Study 40% Syllabus and Score upto 100%
Download E-book

1706449745377

Here, left-hand side is (x+y)^{2} 1706449745230, that is, (x+y)(x+y) 1706449744845

By multiplying the two terms on the left-hand side, we get,

(x+y)(x+y)=x^{2}+xy+xy+y^{2}

1706449744532

(x+y)(x+y)=x^{2}+2xy+y^{2}

1706449743424

Now, this calculated value of left-hand side matches the value on the right-hand side.

Thus, the algebraic identity is proved.


  • (x-y)^{2}=x^{2}-2xy+y^{2}

1706449745599

Here, left-hand side is (x-y)^{2} 1706449743054that is, (x-y)(x-y) 1706449744897

By multiplying the two terms on the left-hand side, we get,

(x-y)(x-y)=x^{2}-xy-xy+y^{2}

1706449742847

(x-y)(x-y)=x^{2}-2xy+y^{2}

Now, this calculated value of left-hand side matches the value on the right-hand side.

Thus, the algebraic identity is proved.1706449743500


  • x^{2}-y^{2}=(x+y)(x-y)

1706449745003

Here, right-hand side is (x+y)(x-y) 1706449743658

By multiplying the two terms on the left-hand side, we get,

(x+y)(x-y)=x^{2}-xy+xy-y^{2}

1706449743756

(x+y)(x-y)=x^{2}-y^{2}

1706449743188

Now, this calculated value of right-hand side matches the value on the left-hand side.

Thus, the algebraic identity is proved.

Frequently Asked Questions (FAQs)

1. Give examples of algebraic identities in three variables.

Two examples of algebraic entities in three variables are as given below,

(x+y+z)^{2}=x^{2}+y^{2}+z^{2}+2xy+2yz+2zx

 x^{3}+y^{3}+z^{3}-3xyz=(x+y+z)(x^{2}+y^{2}+z^{2}-xy-yz-zx)

2. State the various methods used to verify algebraic identities.

There are two simple approaches for verifying algebraic identities: substitution and activity method. In the substitution method, use the arithmetic operation with the replaced values in place of the variables. An additional method for verifying the algebraic identity is the activity methodology.

3. “Every algebraic identity is an equation”. Justify the statement.

A universal equality that holds true for all values is an algebraic identity.All Algebraic identities are algebraic equations which have infinitely many solutions, although not all algebraic equations are identities.

4. Name the two branches of algebra.

The two main branches of algebra are elementary algebra and modern algebra, commonly referred to as abstract algebra.

5. Define algebraic identity.

Algebraic identities are equations in algebra that hold true no matter how each variable is valued.

Articles

Get answers from students and experts
Back to top