Algebra in Maths: Expression, Equation and Sample Questions

Algebra is a broad part of mathematics, together with number systems and geometry. Algebra is the study of mathematical symbols and the rules for handling these symbols. The use of letters in algebra allows us to write formulas in a general manner for particular problems as numbers can be inserted in the place of letters, and thus, we can get solutions to multiple similar type questions with an algebraic expression. 

Definition of a Variable

A variable is a symbol that has no fixed value in any particular expression. It can be represented as many different elements like x,y,l,m,n, a, etc.    

Operations with a variable

Numbers can be added, subtracted, multiplied, and divided with any variable.
Suppose a variable is ‘x.’

Add 10 : x+10

Subtract 10: x-10

Multiply by 10: x*10 

(can also be written as 10x)
Divide by 10: x/10

Rules of Geometry

The perimeter of a Square: Suppose a square has sides ABCD. We know that all sides of a square are identical. Therefore,                  
AB=BC
BC=CD
CD=AD

Let AB = s
Perimeter= Sum of all the sides
     = AB+AD+CD+BC
     = s+s+s+s
     = 4s.

The perimeter of a Rectangle: Suppose a rectangle has sides ABCD. We know that the opposite sides of a rectangle are equal. Therefore,

AB=CD
AD=BC

Let the length AB= l, 
The breadth AD= b 
Perimeter = Sum of all sides
     = AB+BC+CD+AD
     = l+b+l+h
     = 2l + 2b
    = 2(l+b)

Rules of Arithmetic  

Let's understand the different rules of arithmetic briefly:

Commutativity of the addition of two numbers: a+b=b+a

Commutativity of Multiplication of two numbers: a*b=b*a

Distributivity of two numbers: a*(b+c)=a*b+a*c

An algebraic expression is made up of a string of numbers, variables, and operation symbols. This expression represents a value but it does not include 'equal' sign.

Examples:

• 3x + 5
• $7a^2-4ab+b$

Every part of the algebraic expression is separated by an operator. In the expression, variables such as (x,y, a, b) represent unknown numbers. Coefficients (numbers like 3 or 7) represent how many of those variables you have. Unlike an equation, an algebraic expression states a quantity it cannot claim. This quantity is equal to something else until you put “ = ” and another value on other side.

An algebraic equation is a statement where two algebraic expressions are equal. When you put together some numbers, variables, and operation signs on both sides of an equal sign, it becomes an algebraic equation.

3x+5 = 2x−7

 

Here, 3x + 5 must come exactly the same as 2x − 7. The condition here is that x must be the same on both sides. In this case, x = -12.

Following are the branches of algebra. NEET and JEE Main students must learn about these branches in more detail:

Branch	Definition	Example
Elementary algebra	Students have to determine the unknown variable in the expression.	Figure out x in 3x + 5 = 20 ⇒ x = 5.
Linear algebra	In this branch, vector spaces, linear transformations and linear equations are studied.	3 × 3 rotation matrix turns a 3-D model 90° without guessing the angles again and again.
Abstract algebra (groups, rings, fields)	Algebraic structures with their properties and operations are studies in this branch.	RSA encryption depends on “ring” arithmetic with huge mod numbers.
Commutative algebra	It is the same as abstract algebra but here, a b = b a is mandatory.	Does x⁴ + 1 factor over integers uses ideals in ℤ[x].
Algebraic geometry	Treats polynomial equations as geometric objects to study their shapes.	Elliptic curve y² = x³ + ax + b forms the backbone of many crypto-wallets.
Universal algebra	This branch of algebra compares every algebraic structure to find shared patterns.	Shows that both groups and lattices follow the same “homomorphism” rules.

 

Competitive exams such as IIT JAM and IISER often ask application-based questions, and students must be very well-practised in all those topics. Let us consider some sample questions:

1. If there are 50 mangoes in a box, how will you write the total number of mangoes in terms of the number of boxes? (Use b for the number of boxes.) 

Solution.

The total number of boxes = b
Number of mangoes in 1 box = 50
Therefore,
Number of mangoes in b boxes = 50*b = 50b

2. A cube is a three-dimensional figure, as shown in the figure given below. It has six faces, and all of them are identical squares. The length of each side is denoted by s. What will be the total length of edges of the cube?

Solution.

We know that a cube has 12 edges.
Length of each edge of the cube = s

No. of edges of a cube = 12

The total length of the edges of the cube = 12s.

3. Meena, Beena, and Leena are climbing the steps to the hilltop. Meena is at step s, Beena is 8 steps ahead, and Leena 7 steps behind. Where are Beena and Meena? The total number of steps to the hilltop is 10 less than 4 times what Meena has reached. Express the total number of steps using s.

Solution.

According to the question,
Steps of Meena = s
Steps of Beena = Meena’s steps +8
                            = s+8
Steps of Clay = Meena’s steps-7
                        = s-7

Also,
    The total number of steps = 4s-10.