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ALGEBRA : Variable Expressions

Introduction to Variable Expressions

James and Natalie were playing with match sticks and they thought of forming patterns of numbers using the matchsticks.


James took 4 match sticks and formed the number 4

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Natalie added 3 more match sticks to form a pattern with two 4s.

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Then James again added 3 more match sticks to form a pattern with three 4s.

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Suddenly, Natalie got a doubt that how many match sticks are required to make a pattern of ten 4s?


They understood from the existing pattern that they need 4+9(3) sticks to get it done as they want a pattern with ten 4s.


From this, they concluded that they need 4+(n−1)3 sticks, in general, to make a pattern with n number of 4s.


Here, 4+(n−1)3 is called an algebraic expression.


Definition of Variable, Constant, Term and Coefficient

  • A symbol that doesn't have a fixed value is called a variable in Math. It can take any value. In the above example, n is a variable and here it can take the values 1,2,3,...
    Some examples of variables in Math are a,b,x,y,z,m, etc.

  • A symbol that has a fixed numerical value is called a constant. All numbers are constants.
    Some examples of constants are 3,6,−1/2,5 etc.
  • A term is a variable alone (or) a constant alone (or) it can be a combination of variables and constants by the operation of multiplication or division. Some examples of terms are 3x2,−2/3y,5m, etc. Here, the numbers that are multiplying the variables are 3,−2/3 and 5 which are called coefficients.

Variable Expression (Algebraic Expression)

  • A variable expression (or) an algebraic expression is a combination of terms by the operations such as addition, subtraction, multiplication, division, etc.
  • Example of a Variable Expression

An example of a variable expression (or) an algebraic expression is 5X+7

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Evaluating a Variable Expression:


To evaluate a variable expression at a given value, we just substitute that value in the expression and simplify it.


Examples:


Evaluate 5x2+2x+7 at x=−2


Solution:


Substitute x=−2 in 5x^2+2x+7,
5(-2)^2+2(-2)+7 = 5(4)-4+7
=20-4+7
=23

So the answer is 23.


Types of Algebraic expression


There are 3 main types of algebraic expressions which include:

  • Monomial Expression

  • Binomial Expression

  • Polynomial Expression

Monomial Expression:

An algebraic expression which is having only one term is known as a monomial.
Examples of monomial expressions include 3x4, 3xy, 3x, 8y, etc.

Binomial Expression:

A binomial expression is an algebraic expression which is having two terms, which are unlike.
Examples of binomial include 5xy + 8, xyz + x3, etc.

Polynomial Expression:

In general, an expression with more than one term with non-negative integral exponents of a variable is known as a polynomial.
Examples of polynomial expression include ax + by + ca,  x3 + 2x + 3, etc.

Other Types of Expression:


Apart from monomial, binomial and polynomial types of expressions, an algebraic expression can also be classified into two additional types which are:

  • Numeric Expression

  • Variable Expression

Numeric Expression

A numeric expression consists of numbers and operations, but never include any variable. Some of the examples of numeric expressions are 10 + 5, 15 ÷ 2, etc.

Variable Expression

A variable expression is an expression that contains variables along with numbers and operation to define an expression. A few examples of a variable expression include 4x + y, 5ab + 33, etc.


Formulas:


The general algebraic formulas we use to solve the expressions or equations are:

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Quadratic Equation

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