10.1 Introduction

• Simple Algebraic Expressions:

  • Algebraic expressions like $x+3$, $y-5$, $4x+5$, and $10y-5$ are examples of simple expressions.
  • These expressions consist of variables (such as $x$ and $y$) combined with constants (like 3, -5, 4, 5, and -20) using operations like addition and subtraction.

• Usefulness in Formulating Puzzles and Problems:

  • Simple algebraic expressions help in creating and solving mathematical puzzles and problems.
  • They serve as a foundation for understanding more complex algebraic concepts and real-life problem-solving.

• Exposure in Class VII:

  • In earlier classes, particularly in Class VI, students are introduced to the basic concepts of algebra.
  • They learn to recognize and manipulate algebraic expressions, setting the stage for more advanced studies.

• Importance of Expressions in Algebra:

  • Algebraic expressions are a central concept in the study of algebra.
  • Mastery of expressions is essential for progressing to more complex topics like equations, functions, and calculus.

• Goals of This Chapter:

  • Formation of Algebraic Expressions:
    • Understanding how variables and constants are combined using arithmetic operations to form algebraic expressions.
  • Combining Algebraic Expressions:
    • Learning how to add, subtract, multiply, and divide expressions to form new expressions.
  • Finding Values of Expressions:
    • Discovering methods to evaluate expressions for given values of variables.
    • This includes substituting specific numbers for variables and performing the necessary arithmetic operations.
  • Applications of Algebraic Expressions:
    • Exploring how expressions are used in various mathematical contexts, including geometry (e.g., calculating areas and perimeters) and everyday mathematics (e.g., financial calculations, measurements).

10.2 How Are Expressions Formed?

• Variables and Constants:

  • Variables:
    • Represented by letters like $x,y,l,m,$ etc.
    • Variables can take various values; their values are not fixed.
  • Constants:
    • Numbers with fixed values like 4, 100, -17.

• Combining Variables and Constants:

  • Variables and constants are combined using arithmetic operations (addition, subtraction, multiplication, and division) to form algebraic expressions.
  • Examples:
    • $4x+5$: Formed by multiplying the variable $x$ by the constant 4 and then adding the constant 5.
    • $10y-20$: Formed by multiplying $y$ by 10 and then subtracting 20.

• Expressions from Variables and Constants:

  • Expressions are not limited to combinations of variables and constants but can also involve variables combined with other variables.
  • Examples:
    • 𝑥2x2: Formed by multiplying $x$ by itself (𝑥×𝑥=𝑥2x×x=x2).
    • 2𝑦22y2: Formed by multiplying $y$ by itself to get 𝑦2y2, then multiplying 𝑦2y2 by 2.
    • 3𝑥2−53x2−5: Formed by multiplying $x$ by itself to get 𝑥2x2, then multiplying 𝑥2x2 by 3 to get 3𝑥23x2, and finally subtracting 5.
    • $xy$: Formed by multiplying $x$ and $y$ together.
    • $4xy+7$: Formed by multiplying $x$ and $y$ to get $xy$, then multiplying $xy$ by 4 to get $4xy$, and finally adding 7.

• Summary:

  • Forming Expressions:
    • Algebraic expressions are formed by combining variables and constants using arithmetic operations.
  • Operations:
    • Addition, subtraction, multiplication, and division are used to create expressions from variables and constants.
  • Examples of Complex Expressions:
    • Expressions can involve variables multiplied by themselves, by constants, or by other variables.
  • Understanding Terms:
    • Each part of an expression formed separately is called a term.
    • For example, in $4x+5$, $4x$ and 5 are terms.
    • Terms are combined to form algebraic expressions.

10.3 Terms of an Expression

• Definition of Terms:

  • Terms: Parts of an algebraic expression that are added together.
  • Formation: Each term is formed separately and then combined to form an expression.

• Examples of Terms:

  • Expression: $4x+5$
    • Terms: $4x$ and $5$
  • Expression: 3𝑥2+7𝑦3x2+7y
    • Terms: 3𝑥23x2 and $7y$

• Identifying Terms:

  • Terms are separated by addition (+) or subtraction (−) signs.
  • Example: In 4𝑥2−3𝑥𝑦4x2−3xy, the terms are 4𝑥24x2 and $-3xy$.

• Understanding Factors:

  • Factors of a Term: Numbers and variables that multiply together to form a term.
  • Examples:
    • 4𝑥24x2 has factors 4, $x$, and $x$.
    • $-3xy$ has factors -3, $x$, and $y$.

• Tree Diagrams:

  • Tree diagrams can help visualize the terms and their factors.
  • Example:
    • Expression: 4𝑥2−3𝑥𝑦4x2−3xy
      • Term: 4𝑥24x2
        • Factors: 4, $x$, $x$
      • Term: $-3xy$
        • Factors: -3, $x$, $y$

• Coefficients:

  • Definition: The numerical factor of a term.
  • Examples:
    • In $5xy$, 5 is the coefficient.
    • In −7𝑥2𝑦2−7x2y2, -7 is the coefficient.

• General Use of Coefficients:

  • Sometimes, a coefficient can be an algebraic factor or a product of several factors.
  • Example:
    • In $5xy$, 5 is the coefficient of $xy$.
    • In 10𝑥𝑦210xy2, 10 is the coefficient of 𝑥𝑦2xy2.

• Identifying Terms and Coefficients:

  • Example 1: $xy+4$
    • Term: $xy$ – Coefficient: 1
  • Example 2: 13−𝑦213−y2
    • Term: −𝑦2−y2 – Coefficient: -1
  • Example 3: 13−𝑦+5𝑦213−y+5y2
    • Term: $-y$ – Coefficient: -1
    • Term: 5𝑦25y2 – Coefficient: 5
  • Example 4: 4𝑝2𝑞−3𝑝𝑞2+54p2q−3pq2+5
    • Term: 4𝑝2𝑞4p2q – Coefficient: 4
    • Term: −3𝑝𝑞2−3pq2 – Coefficient: -3

• Practice Identifying Coefficients:

  • Expressions and Coefficients:
    • $4x-3y$: Coefficient of $x$ is 4
    • $8-x+y$: Coefficient of $x$ is -1
    • 𝑦2𝑥−𝑦y2x−y: Coefficient of $x$ is 𝑦2y2
    • $2z-5xz$: Coefficient of $x$ is -5z
    • $4x-3y$: Coefficient of $y$ is -3
    • $8+yz$: Coefficient of $y$ is $z$
    • 𝑦𝑧2+5yz2+5: Coefficient of $y$ is 𝑧2z2
    • $my+m$: Coefficient of $y$ is $m$

10.4 Like and Unlike Terms

• Definition:

  • Like Terms: Terms that have the same algebraic factors.
  • Unlike Terms: Terms that have different algebraic factors.

• Example of Like Terms:

  • In $2xy-3x+5xy-4$:
    • $2xy$ and $5xy$ are like terms because they both have the factors $x$ and $y$.
  • Similarly, $-3x$ and $-4$ are like terms because they both have no variables.

• Example of Unlike Terms:

  • In $2xy-3x+5xy-4$:
    • $2xy$ and $-3x$ are unlike terms because they have different algebraic factors.
    • Likewise, $2xy$ and $-4$ are also unlike terms.

• Understanding Factors for Comparison:

  • Factors such as variables and their powers are compared to determine if terms are like or unlike.
  • Coefficients (the numerical part of a term) are not considered when determining likeness.

• Importance in Simplifying Expressions:

  • Identifying like terms is crucial when simplifying expressions through addition or subtraction.
  • Like terms can be combined by adding or subtracting their coefficients while keeping their variables unchanged.

• Example:

  • In $3x+2y-5x+4y$:
    • $3x$ and $-5x$ are like terms because they both have the factor $x$.
    • $2y$ and $4y$ are also like terms because they both have the factor $y$.
    • Therefore, the expression can be simplified to $(3x-5x)+(2y+4y)$, which becomes $-2x+6y$.

• Practice Identifying Like and Unlike Terms:

  • In $2xy-3x+5xy-4$:
    • $2xy$ and $5xy$ are like terms.
    • $-3x$ and $-4$ are like terms.
    • But $2xy$ and $-3x$ are unlike terms.

• General Rule:

  • To determine if terms are like or unlike, compare the variables and their powers.
  • If the variables and their powers are the same, the terms are like; otherwise, they are unlike.

10.5 Monomials, Binomials, Trinomials, and Polynomials

• Definition:

  • Monomial: An algebraic expression with only one term.
  • Binomial: An algebraic expression with two unlike terms.
  • Trinomial: An algebraic expression with three unlike terms.
  • Polynomial: An algebraic expression with one or more terms.

• Examples:

  • Monomial: $4xy$, −5𝑎2𝑏−5a2b
  • Binomial: $x+y$, $3m-2n$
  • Trinomial: 2𝑥2−𝑥𝑦+12x2−xy+1, 𝑎3+𝑎𝑏−3𝑎a3+ab−3a

• Understanding Terms:

  • Monomial: Consists of one term. Example: $4xy$
  • Binomial: Consists of two unlike terms. Example: $x+y$
  • Trinomial: Consists of three unlike terms. Example: 2𝑥2−𝑥𝑦+12x2−xy+1
  • Polynomial: Consists of one or more terms. Example: 4𝑥2+3𝑥𝑦−2𝑦4x2+3xy−2y

• Classification:

  • Based on Number of Terms:
    • Monomial: 1 term
    • Binomial: 2 terms
    • Trinomial: 3 terms
    • Polynomial: More than 3 terms

• Examples of Polynomials:

  • 3𝑥2+2𝑥𝑦−5𝑦3x2+2xy−5y
  • 𝑎3+4𝑎2𝑏−2𝑎𝑏2+𝑏3a3+4a2b−2ab2+b3

• Importance in Algebra:

  • Polynomials are fundamental in algebraic expressions and equations.
  • They are used to model various real-world situations and solve mathematical problems.

• Practice Identifying:

  • Monomial: 5𝑥25x2, $-3ab$
  • Binomial: $x-y$, $2m+3n$
  • Trinomial: 2𝑥2−𝑥𝑦+12x2−xy+1, 𝑎3+𝑎𝑏−3𝑎a3+ab−3a
  • Polynomial: 4𝑥2+3𝑥𝑦−2𝑦4x2+3xy−2y, 𝑥4−2𝑥2+3𝑥−7x4−2x2+3x−7

• Understanding Algebraic Expressions:

  • Polynomials are used to represent relationships and solve equations in various fields, including physics, engineering, and economics.

10.6 Finding the Value of an Expression

• Definition:

  • Finding the value of an algebraic expression involves substituting specific values for the variables in the expression and evaluating the result.

• Importance:

  • Essential in various mathematical contexts, including solving equations, evaluating formulas, and solving real-world problems.

• Example:

  • Consider the expression $3x+2y$:
    • If $x=4$ and $y=5$, we substitute these values into the expression:
      • $3(4)+2(5)=12+10=22$
    • Therefore, when $x=4$ and $y=5$, the value of the expression $3x+2y$ is 22.

• Applications:

  • Equation Solving: Used to verify if a particular value of a variable satisfies an equation.
  • Geometry: Used in formulas to find the area, perimeter, volume, etc., of geometric shapes.
  • Real-world Problems: Used to solve problems involving quantities that vary, such as cost, distance, time, etc.

• Steps:

  1. Substitute: Replace each variable in the expression with the given value.
  2. Evaluate: Perform the arithmetic operations to find the result.

• Example Problem:

  • Problem: Find the value of the expression 2𝑥2−3𝑥𝑦+4𝑦2x2−3xy+4y when $x=3$ and $y=2$.
  • Solution:
    • Substitute $x=3$ and $y=2$ into the expression:
      • 2(3)2−3(3)(2)+4(2)2(3)2−3(3)(2)+4(2)
    • Evaluate each term:
      • $2(9)-3(6)+8=18-18+8=8$
    • Therefore, when $x=3$ and $y=2$, the value of the expression is 8.

• Practice:

  • Given an expression and specific values for the variables, students can practice finding the value of the expression by substitution and evaluation.

• Real-world Examples:

  • Geometry: Finding the area of a rectangle given its length and width.
  • Physics: Using formulas to calculate velocity, acceleration, force, etc.
  • Finance: Calculating interest, discounts, profits, etc., in business applications.