Introduction to Algebra

Introduction 

What is Algebra?

Algebra is the branch of Maths that uses alphabetical letters to find unknown numbers. These letters are also called variables. The values which are known in the given expression such as numbers are called constants.

Algebra involves the simple operation of mathematics like addition, subtraction, multiplication, and division involving both constant as well as variables. For example, x+10 = 0.

Introduction to Algebra-PDF

Algebra involves the simple operation of mathematics like addition, subtraction, multiplication, and division involving both constant as well as variables. Here is a downloadable PDF to explore more.

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Introduction to Algebra-PDF

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Also read:

• Algebra in Real Life
• Algebra vs Calculus   
• The History of Algebra
• Algebra in Real Life
• How to factorize a polynomial? Part-1 

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What are algebraic Expressions?

Algebraic expressions in maths are made up of variables and constants along with operations like addition, subtraction, multiplication, and division.

These expressions can also be named as Terms, which again also called Algebraic Equations. The values which are known in the given expression are represented with the help of variables, constants, and coefficients. 

Algebra Expressions covers the simple operation of mathematics like; for example x+10.

Terms related to basic algebra skills are mentioned below.

• Exponent

• Expression

• Polynomial (Monomial, binomial and trinomial)

• Like terms and Unlike terms

• Constants

1. An equation is a statement which implies two same identities separated by the “=” sign. 

2. An expression is a group of different terms separated by ‘+’ or ‘-‘ sign. 

3. Like terms are those terms whose variables and their exponents are the same. 

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Algebraic Identities

Following are the algebraic identities:

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

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Properties of Algebra

Algebra follows the same set of rules as Number Systems. The order of operations is preserved, and the laws governing addition and multiplication still apply.

However, with the inclusion of variables comes a new set of challenges. For a definitive value for an expression, we need to evaluate an algebraic expression for specific values of the variables, observe how the value of the expression changes as the values of the variables changes.

Like operations with numbers, operations with variables obey certain laws.

The commutative law tells us that we can change the order of the terms while performing addition or multiplication.

• a+b=b+a
• a*b=b*a

The associative law tells us that in addition or multiplication, we can associate the terms or factors as we are comfortable with. 

• (a+b)+c=a+(b+c)

Both the commutative law and the associative law apply to either addition or multiplication, but not with both operations together.

The distributive law deals with the combination of addition and multiplication. When a sum is multiplied by value, the value is distributed to each part of the sum.

• a(b+c)=a(b)+a(c)

While solving questions in algebra follow the below steps:

• Identify the relationship between expressions. 

• If an expression has a known value, write the other expression in terms of the expression with a known value. 

• Substitute the known value for the expression.

• Evaluate the expression.

For Example: 

\(2(3x - 7) + 4 (3 x + 2) = 6 (5 x + 9 ) + 3\)  
\(6x-14+12x+8=30x+54+3\)
\(18x - 6=30x + 57\)
\(-6=12x +57\)
\(-63=12x\)
\(x= - 63/12\)
\(x= - 21/4\) 

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Example Problems

1. Add \(6x + 4y – z + 3\), \(2y – 3z + 4\), \(y – 7x + 2z – 1\) and \(2z – 5x – 6\).
(A) \(4x – 6y + 2\)     (B) \(–3x + 14y – 3z + 2\)    (C) \(–6x + 7y\)     (D) \(–6x + 6y + z – 4\)

Answer:

(C) \(–6x + 7y\)

2. 

(A) 5     (B) 7     (C) 9     (D) 11

 

Answer:

(B) 7

3. Simplify the expression: \(a – 2b – [5a – 6b – {3a – c + (5a – 2b – 3a – c)}]\)
(A) \( 3a – 2a + 2c\)     (B) \(a + 2b – 2c\)     (C) \( 4a + 6b\)     (D) \(–2b +2c\)

 

Answer:

(B) \(a + 2b – 2c\)

4. \(3l (l – 4m + 5n)\) and \(4l (10n – 3m + 2l)\) are algebraic expressions. What is the algebraic expression obtained when subtracting the first expression from the second expression?
(A) 25lm + 5l²     (B) 25ln – 5l²     (C) 25ln + 5l²     (D) 25lm – 5l²

 

Answer:

(C) 25ln + 5l2

5. Factorize (2a + 3b)² – (3a – 2b)².
(A) (5a + b) (5a – b)                      (B) (a + 5b) (a – 5b)
(C) (5a + b) (5b – a)                      (D) (5a + b) (5b + a)

 

Answer:

(C) (5a + b) (5b – a)

6.

(A) 1        (B) 0        (C) 3       (D) 2

Answer:

(D) 2

7. The perimeter of a triangle is 9x² – 4x + 3 and the length of two of its sides are 5x² – 2x + 4 and 2x² + 2x – 3. Find the length of the third side of the triangle.
(A) 2x² + 4x – 2                  (B) 2x² – 4x + 2
(C) 2x² + 4x – 2                  (D) 2x² – 4x – 2

 

Answer:

(B) 2x2 – 4x + 2

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Summary

So here we come to the end of this article, let us recapitulate what we learned, So we learned about:

• How to Simplify Algebraic Equations using the BODMAS rule. 

• Performing various operations on algebra. 

• Solving word problems by reading the statements carefully and writing them down step by step.

Hope you all got relevant information on the topic. You all are free to write comments and like this article. 

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