Algebraic Expressions – Notes for CBSE Class 7 Maths

Introduction

Algebraic expressions are a fundamental part of mathematics that combine numbers, variables, and operations (like addition, subtraction, multiplication, and division). In this chapter, students learn how to form, interpret, and simplify algebraic expressions, which serve as the building blocks for solving equations and understanding patterns in higher classes.

1. What are Algebraic Expressions?

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. Variables are letters (like $x$, $y$, or $z$) that represent unknown or changing values, while numbers are constants.

• Examples:
  • $3x+5$
  • $2y-7$
  • $4a+3b-9$

Unlike numerical expressions (e.g., $5+3=8$), algebraic expressions do not give a fixed value because the variables can take any value.

2. Terms, Factors, and Coefficients

An algebraic expression is made up of terms, which are separated by plus (+) or minus (-) signs.

• Terms: Individual parts of an expression.
  • In $3x+5y-7$, the terms are $3x$, $5y$, and $-7$.
• Factors: Numbers or variables that multiply together to form a term.
  • In $3x$, the factors are $3$ and $x$.
• Coefficient: The numerical factor of a term that contains a variable.
  • In $3x$, the coefficient is $3$.
  • In $-5y$, the coefficient is $-5$.
• Constant Term: A term with no variable, like $-7$ in the expression above.

3. Types of Algebraic Expressions

Based on the number of terms, algebraic expressions are classified as:

1. Monomial: An expression with one term.
   • Examples: $5x$, $-3y$, $7$.
2. Binomial: An expression with two terms.
   • Examples: $2x+3$, $4a-5b$.
3. Trinomial: An expression with three terms.
   • Examples: $3x+2y-1$, $5a-4b+7$.
4. Polynomial: An expression with one or more terms (monomial, binomial, trinomial, etc. are all polynomials).

4. Like and Unlike Terms

• Like Terms: Terms that have the same variable raised to the same power.
  • Examples: $3x$ and $5x$, 2y22y^22y2 and 7y27y^27y2.
• Unlike Terms: Terms that have different variables or different powers of the same variable.
  • Examples: $3x$ and $4y$, $2x$ and 5x25x^25x2.

Only like terms can be added or subtracted directly.

5. Operations on Algebraic Expressions

(i) Addition and Subtraction

• Combine like terms by adding or subtracting their coefficients.
• Unlike terms remain separate.

Example:

• Add $3x+5y-2$ and $2x-3y+4$:
  • Step 1: Group like terms:
    • $3x+2x=5x$
    • $5y-3y=2y$
    • $-2+4=2$
  • Step 2: Write the result: $5x+2y+2$.

(ii) Multiplication

When multiplying algebraic expressions:

• Multiply the coefficients and add the exponents of the same variable (if any).
• Use the distributive property when multiplying a monomial with a binomial or polynomial.

Example:

• Multiply $2x$ and $3x$:
  • 2x×3x=(2×3)×(x×x)=6x22x \times 3x = (2 \times 3) \times (x \times x) = 6x^22x×3x=(2×3)×(x×x)=6x2.
• Multiply $3x$ and $2x+5$:
  • 3x×(2x+5)=3x×2x+3x×5=6×2+15x3x \times (2x + 5) = 3x \times 2x + 3x \times 5 = 6x^2 + 15x3x×(2x+5)=3x×2x+3x×5=6x2+15x.

6. Finding the Value of an Expression

To find the value of an algebraic expression, substitute the given values of the variables and simplify.

Example:

• For $2x+3y$, if $x=4$ and $y=2$:
  • Substitute: $2(4)+3(2)=8+6=14$.

7. Formation of Algebraic Expressions

Algebraic expressions can represent real-life situations or patterns.

• Example: If the cost of one pen is $x$ rupees and one pencil is $y$ rupees:
  • Cost of 3 pens and 2 pencils = $3x+2y$.

8. Simplification of Algebraic Expressions

Simplification involves combining like terms and performing operations.

Example:

• Simplify $4x+3y-2x+5y-7$:
  • Combine like terms:
    • $4x-2x=2x$
    • $3y+5y=8y$
    • Constant: $-7$
  • Result: $2x+8y-7$.

9. Key Points to Remember

• A variable can take any value, while a constant has a fixed value.
• The degree of a monomial is the exponent of the variable (e.g., x2x^2x2 has degree 2).
• The degree of a polynomial is the highest degree of its terms.
• Always check for like terms before adding or subtracting.

10. Examples for Practice

1. Identify the terms, coefficients, and constants in $5x-3y+8$.
   • Terms: $5x$, $-3y$, $8$
   • Coefficients: $5$ (of $x$), $-3$ (of $y$)
   • Constant: $8$
2. Add $7x-4y+3$ and $2x+5y-1$.
   • $7x+2x=9x$
   • $-4y+5y=y$
   • $3-1=2$
   • Result: $9x+y+2$
3. Multiply $4x$ and $3x-2$.
   • 4x×3x+4x×(−2)=12×2−8x4x \times 3x + 4x \times (-2) = 12x^2 – 8x4x×3x+4x×(−2)=12x2−8x.

11. Applications of Algebraic Expressions

• Patterns: Algebraic expressions help describe patterns.
  • Example: The number of squares in the $n$-th figure of a pattern might be $2n+1$.
• Word Problems: Translate real-life situations into expressions.
  • Example: If a book costs $p$ rupees and a notebook costs $q$ rupees, the total cost of 2 books and 3 notebooks is $2p+3q$.

12. Common Mistakes to Avoid

• Do not add or subtract unlike terms (e.g., $3x+4y$ cannot be simplified further).
• Be careful with signs when combining terms (e.g., $5x-(-3x)=5x+3x=8x$).
• Ensure proper use of the distributive property in multiplication.

13. Summary

Chapter 12 introduces students to the world of algebra by explaining how to form and manipulate algebraic expressions. Key skills include identifying terms, performing operations (addition, subtraction, multiplication), and evaluating expressions by substituting values. Mastery of these concepts lays the foundation for solving equations and understanding more complex algebraic topics in future classes.

Download pdf notes and Sulutions of the Chapter:

Chapter 13: Exponents and Powers

Please Visit Readspot for Hindi Medium Study Material