Key Concepts and Examples of Variable Expressions
Algebra is the branch of mathematics that deals with symbols and variables. Alphabetical letters are used to find unknown numbers from the equations. Algebra is divided into sub-branches such as elementary algebra, advanced algebra, linear algebra, and commutative algebra. It covers algebraic expressions, formulas, and identities, which are used in solving many mathematical problems. An algebraic expression is an expression consisting of variables, constants, coefficients, and mathematical operations like addition, subtraction, etc. An algebraic expression is a form of writing equations using letters or alphabets without specifying their values. These letters are called variables. In this article, we will learn about expressions with variables.
Parts of Variable Expressions
A variable expression is a combination of different terms involving variables, constants, and mathematical operations like addition, subtraction, etc.
Some of the important parts of an algebraic expression are defined below:
Variable: A symbol without a fixed value is called a variable. It can take any value. It is represented by alphabetical letters like x, y, z, etc.
Constant: A symbol with a fixed numerical value is called a constant.
Term: A term is a variable or a constant alone or a combination of both combined by mathematical operations.
Coefficients: The quantity multiplied by a variable and remains constant throughout the problem is known as a coefficient.
Algebraic expression example: 9x + 7.
Here x is the variable, 9 is the coefficient 9x, 7 is the constant, and 9x and 7 are the two terms in the expression.
Types of Algebraic Expressions
There are three main types of algebraic expressions based on the number of terms:
Monomial expression: An algebraic expression with only one term is known as a monomial expression.
For example – 3x, 4, 8y, etc
Binomial expression: An algebraic expression with only two terms is known as a binomial expression.
For example – $ax + by$, \[x^2+y\], etc
Trinomial expression: An algebraic expression with three terms is known as a trinomial expression.
For Example: $7x+4y-3$, \[x^2+3y -8\], etc.
Polynomial expression: An algebraic expression having one variable and the exponent of the variable is a whole number, is known as a polynomial expression.
For example – $ax + bx + cx+6$, \[x^4+ x^2 – x + 4\] etc
Use of Algebraic Expressions
Some of the uses of algebraic expressions are listed below:
They are used to solve different and complex equations in mathematics.
They can also be seen in computer programming.
They are also used in economics to find out the revenue, cost, etc.
Different branches of mathematics like trigonometry, geometry, etc also use algebraic expressions to find the unknown values of angles.
Algebraic expressions are also used to represent real-life problems.
Algebraic Expression Formulas
The basic identities used in algebra are also known as algebraic expression formulas:
\[(a+b)^2=a^2+2ab+b^2\]
\[(a−b)^2=a^2−2ab+b^2\]
\[(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc\]
\[a^2−b^2=(a+b)(a−b)\]
\[(a+b)^3=a^2+3a^2b+3ab^2+b^3\];\[(a+b)^3=a^3+b^3+3ab(a+b)\]
\[(a−b)^3=a^3−3a^2b+3ab^2−b^3; (a−b)^3=a^3−b^3−3ab(a-b)\]
\[a^3+b^3=(a+b)(a^2−ab+b^2)\]
\[a^3−b^3=(a−b)(a^2+ab+b^2)\]
Interesting facts
The terms in algebraic expressions that are constants or involve the same variables raised to the same exponents are called like terms. For example: \[6x^2 + x – 4x^2 + 9\], here \[6x^2\] and \[– 4x^2\] are similar terms.
The terms in algebraic expressions that do not have the same variables or have the same variables but are raised to different exponents are called, unlike terms. For example: \[6x^4 + x – 4x^2 + 9\], here \[6x^2\], x and 9 are the unlike terms.
All the polynomials are algebraic expressions but all algebraic expressions are not polynomials.
The algebraic expressions which do not have fractional or non–negative exponents are polynomials.
Solved questions
Q1. Name the type of algebraic expression:
2x + 3y + 24xy
– 3x + 2y
– 20xy
Ans. a. Trinomial b. Binomial c. Monomial
Q2. Evaluate the expression \[6ab + 2b^2 – c\], where a = 2, b = 3, c = 1.
Ans. Put the values of a, b and c is the expression
\[6(2)(3) + 2(3)^2 – 1 = 53\]
Q3. Evaluate \[{(a + b)}^{2}\] and verify the identity for a = 2 and b = 1.
Ans. We know,\[(a+b)^2=a^2+2ab+b^2\]
LHS \[= (a+b)^2 = (2+1)^2 = 9\]
RHS \[= a^2+2ab+b^2 = 2^2 + 2(2)(1) + 1^2 = 4 + 4 +1 = 9\]
Summary
Algebra is the branch of mathematics that deals with symbols and variables. A variable expression is a combination of different terms involving variables, constants, and mathematical operations like addition, subtraction, etc. There are three main types of algebraic expressions based on the number of terms: Monomial, Binomial, and Polynomial.
Practice questions
Q1. Name the type of algebraic expression:
\[{2x}^{2} \]
– 3x + y
\[– 20xy + 3 – {2y}^2\]
Ans. a. monomial b. binomial c. trinomial
Q2. Evaluate the expression ab + 2b – c, where a = 1, b = 2, c = 3.
Ans. 3
Q3. Evaluate ${(a-b)}^2$ and verify the identity for a = 1 and b = - 1.
Ans. 4
List of related articles
FAQs on Mastering Expressions with Variables
1. What is the main difference between a variable and a constant in an expression?
The main difference lies in their value. A constant is a fixed numerical value that does not change, such as 7, -15, or π. A variable is a symbol, typically a letter like 'x' or 'y', used to represent a quantity that can change or is unknown. In the expression 2x + 7, 'x' is the variable and '7' is the constant.
2. What are expressions with variables, and how are they formed?
An expression with variables, also known as an algebraic expression, is a mathematical phrase made by combining numbers, variables, and at least one operation sign (+, -, ×, ÷). For example, to form the expression '5 less than the product of x and 3', you would write it as 3x - 5. It doesn't have an equals sign, so it represents a value that changes as the variable changes.
3. What are some examples of different types of algebraic expressions based on their terms?
Algebraic expressions are often classified by the number of terms they contain. Terms are the parts of the expression separated by addition or subtraction signs. The main types are:
- Monomial: An expression with a single term, like 8xy or -4a².
- Binomial: An expression with two unlike terms, such as 3x + 9 or a² - b².
- Trinomial: An expression with three unlike terms, such as x² + 5x - 6.
Expressions with one or more terms, where the variables have whole number exponents, are also called polynomials.
4. How do you evaluate an expression with variables?
To evaluate an expression, you must be given a specific numerical value for each variable. You substitute this value into the expression and then perform the calculations following the order of operations (BODMAS/PEMDAS). For instance, to evaluate the expression 4a - 1 when a = 3, you replace 'a' with 3 to get 4(3) - 1, which simplifies to 12 - 1 = 11.
5. What is the fundamental difference between an algebraic expression and an equation?
The key difference is the presence of an equals sign (=). An expression, like 5x + 2, is a phrase representing a value that can change depending on x. You can simplify or evaluate it. An equation, like 5x + 2 = 12, is a complete mathematical sentence stating that two quantities are equal. You solve an equation to find the specific value of the variable that makes the statement true.
6. Are all algebraic expressions also polynomials?
No, not all algebraic expressions are polynomials. A polynomial is a specific type of algebraic expression where the variables have only non-negative whole number exponents (e.g., 0, 1, 2, 3...). An expression like 3√x + 5 or 7/y - 2 is an algebraic expression, but it is not a polynomial. This is because it involves a fractional exponent (√x is x1/2) or division by a variable (7/y is 7y-1). Therefore, all polynomials are algebraic expressions, but not all algebraic expressions are polynomials.
7. Why are expressions with variables important in real-life applications?
Expressions with variables are essential for creating formulas and models for real-world situations where values are not fixed. For example, a taxi fare can be represented by the expression 50 + 15k, where 50 is the initial charge and 'k' is the variable for kilometres travelled. They form the foundation for solving problems in science (like F = ma), finance (calculating interest), and engineering (designing structures).
