Constant and Variable in Algebraic Expressions - Definition and Examples
Algebra is a subpart of mathematics that deals with relations, operations, and their constructions. It is one of the building blocks of mathematics, and it finds a huge variety of applications in our day-to-day life.
Algebra is one of the many various branches of Mathematics. It deals with the binary relations, operations and constructions of various mathematical functions. Algebra is an integral part of a student’s education. It helps in developing the overall understanding of other Math branches. The other branches of Math include Calculus, Arithmetic, and Geometry etc.
The word Algebra is taken from the Arabic phrase al-jebr which means "reunion of broken parts". It is the study of multiple operations and relations, various constructions of geometric figures and the explanation and proofs of concepts such as polynomials, equations, and algebraic structures.
Now, the question arises, how is Algebra different from the other branches in Math? The answer to this is the Algebraic Expressions used in Algebra.
To know further, read on.
What is an Algebraic Expression?
The expression of numbers using alphabets or letters with their actual values unknown are known as algebraic expressions. An algebraic expression is one or more algebraic terms used to express a concept. The basic building blocks of an Algebraic expression are variables and constants and operating symbols such as plus, minus signs.
3x2 + 2y + 7xy + 5
Algebraic expressions are just a set of variables and constants that are separated by plus or minus signs.
In this article, we will mainly focus on the definition and properties of constants and variables.
Algebraic expressions can be monomial, binomial and polynomial. Given below are the definitions of the three types.
Monomial Expression
Algebraic Expressions having one term is known as a monomial.
7xy is an example of a monomial Expression.
Binomial Expression
Algebraic Expressions are usually having w=two terms are called binomial.
4x+6 is an example of a binomial expression.
Polynomial Expression
This expression generally contains more than two terms with non-negative integral components of variables is known as a Polynomial.
ax+3y+6c=0 is an example of a polynomial expression.
There are two types of Expressions; they are Numeric Expressions and Variable Expressions.
Numeric Expression consists of numbers; never include variables.
For example,
10-2, 3*4 etc.
Variable Expressions contain both variables and numbers to define a particular expression.
For example,
5x+y, 5ac-44 etc.
What are Constants?
Constants are that part of the algebraic expression that involve only numbers. We call them constants because the value is always the same. It is definite. There are no variables in the term that can change the value of the constant. The number 8 is a constant, because it has a particular value, and it is known to everyone. It can’t be changed. In the expression 3XY – 4 = 2Y, the second term 4 is the constant. 3XY cannot be a constant because the values X and Y can be varied, and hence will vary the entire term 3XY.
Constants are the symbols that have fixed numerical values such as 2,8,-5,-7,12, etc.
What are Variables?
In an algebraic expression, the letters represent variables. They are not constant; the value of these terms can be changed from time to time. In the expression 2y+3x=0, the values x and y represent variables. Y can be 2,3, or any number and x can also be any number that satisfies the equation. More than one value can be substituted for the letters in an algebraic expression.
Variables are quantities whose values are unknown which means they have no numerical values and are represented by an alphabet such as a,b,c,d, etc.
Constants and variables are a part of algebraic expressions and are considered the basic building blocks of algebraic expressions. For example-
For an algebraic expression
2x+3
3 is a constant since it has a fixed value
x is a variable, while 2 is the coefficient of x
Some more Examples of Variables and Constants
In 4a, a is variable and 4 is a constant.
In -5pq, p and q are variables and -5 is a constant.
In 8y, y is a variable and 8 is a constant but 8y together is a variable.
The product of variables and constants is always a variable.
Difference between Variables and Constants
The Entire Article in a Nutshell
Consider the type statement ‘4x+2y=12.’
What does ‘x’ and ‘y’ represent? They are quantities that can be continuously varied and can be substituted with different values. Hence, it is called a variable. It is usually an English alphabet that represents these values. A variable is a letter that signifies an unidentified. It always serves a number, but it carries varying values when written in the expression. In this expression x and y are variables. All algebraic expressions and terms consist of a minimum of one variable. It is this variable that distinguishes an algebraic expression from an arithmetic one. The variable in a mathematical expression can explain infinite possibilities to determine the value of the expression.
The constants over here are 4, 2 and 12. These values cannot be changed and represent a fixed mathematical expression at all times. Under no circumstances can the values be assumed differently or written differently.
Example 1. Substitute the variables and constants from the following expressions.
2
3y+5
ax2+bx+c2
\[\sqrt{2}\] +xa
\[\frac{1}{\sqrt{2+x}}\]
Ans:
Example 2. State whether the following statements are true or false.
10 is a constant, z is a variable but 10z is a variable.
2 is a constant, p is a variable but 2+p is a variable.
12 is a constant, m is a variable but 12+m is a constant.
0 is a constant.
Ans:
True
True
False
True
Formulas used in Algebraic Equations
(a+b)2=a2+2ab+b2
(a-b)2=a2-2ab+b2
a2-b2=(a+b)(a-b)
(a+b)3=a3+b3+3ab(a+b)
(a-b)3=a3-b3-3ab(a-b)
a3+b3=(a+b)(a2-ab+b2)
a3-b3=(a-b)(a2+ab+b2)
FAQs on Variables and Constants in Algebraic Expressions
1. What are variables and constants in an algebraic expression?
In algebra, a variable is a symbol, usually a letter (like x, y, or a), that represents an unknown value that can change. A constant is a fixed value that does not change. For example, in the expression 2x + 5, 'x' is the variable because its value can vary, while '5' is the constant because its value is always five.
2. How can you identify the variables, constants, and coefficients in an expression like 7a - 4b + 9?
To identify the parts of the expression 7a - 4b + 9, you can break it down as follows:
- Variables: The letters representing unknown values, which are a and b.
- Constants: The term with a fixed value that does not have a variable attached. Here, the constant is +9.
- Coefficients: The numbers multiplied by the variables. The coefficient of 'a' is 7, and the coefficient of 'b' is -4.
3. What is the difference between a monomial, a binomial, and a polynomial?
The difference lies in the number of terms they contain:
- A monomial is an algebraic expression with only one term (e.g., 5x, 8, or 2xy).
- A binomial is an algebraic expression with two unlike terms (e.g., 5x + 3, or a - 2b).
- A polynomial is a general term for an algebraic expression with one or more terms. Monomials and binomials are both types of polynomials. An expression with three terms, like x² + 3x - 4, is often called a trinomial.
4. Why do we use letters as variables in Maths instead of just using numbers?
We use letters as variables in Maths to represent quantities that are unknown or can change. This is powerful because it allows us to:
- Generalise relationships: For example, the formula for the area of a rectangle, A = l × w, works for any length (l) and width (w).
- Solve for unknowns: Variables let us set up equations to find specific missing values in a problem.
- Model real-world situations: They help describe scenarios where values change, like calculating the total cost based on the number of items purchased.
5. In a term like 5y, is the whole term '5y' considered a variable or a constant?
This is a common point of confusion. While 'y' is the variable and '5' is the coefficient, the entire term '5y' is considered a variable term. This is because its total value depends on the value of 'y'. If 'y' changes, the value of '5y' also changes. A term is only a constant if it contains no variables at all, such as '8' or '-15'.
6. Can you provide a real-world example to explain variables and constants?
Certainly. Imagine ordering a pizza. There is a fixed delivery fee of ₹50, and each pizza costs ₹400. We can create an algebraic expression for the total cost: Total Cost = 400p + 50.
- Here, p (the number of pizzas) is the variable because it can change depending on how many pizzas you order.
- The delivery fee, ₹50, is the constant because it remains the same regardless of how many pizzas you buy.
7. What is the main difference between a 'term' and a 'coefficient' in an algebraic expression?
A term is a single component of an expression, which can be a number, a variable, or a product of numbers and variables (like 7x, 3y, or -10). In the expression 7x + 3y - 10, the terms are 7x, 3y, and -10. A coefficient is specifically the numerical factor that is multiplied by a variable within a term. So, in the term '7x', the coefficient is 7.
