Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 → Algebra ↓


Algebraic Expressions


Algebraic expressions play an important role in mathematics, especially in Class 8 where the basic concepts are introduced. Understanding algebraic expressions is the key to understanding more complex mathematical ideas. This lesson will provide a detailed explanation of algebraic expressions, including examples, structure, and the various operations you can perform with them.

What is an algebraic expression?

In simple terms, an algebraic expression is a mathematical phrase that can contain numbers, variables (such as letters) and operations such as addition, subtraction, multiplication and division. Unlike equations, expressions do not have an equal sign (=).

For example:

3x + 4

This expression contains a variable x, a coefficient 3, and a constant 4.

Components of an algebraic expression

An algebraic expression has different components:

1. Variables

Variables are symbols used to represent unknowns. They are usually represented by letters such as x, y, z, etc. In the expression 5x + 7, x is the variable.

2. Constants

Constants are fixed values that do not change. In the above expression, 7 is a constant.

3. Coefficient

Coefficients are numbers that multiply a variable. They are placed right in front of the variable. In 5x, the number 5 is the coefficient.

4. Terms

A term can be a single number, a single variable, or a combination of numbers and variables. The expression 3x + 4 has two terms: 3x and 4.

5. Operator

These are symbols that represent operations. Common operators in algebraic expressions include + (addition), - (subtraction), * (multiplication) and / (division).

Basic examples and structure

To understand algebraic expressions better, let's look at some examples:

Example 1:

4x + 5y - 9

This expression has three terms: 4x, 5y, and -9. Here, x and y are variables; 4 and 5 are coefficients; -9 is a constant. The operators are + and -.

Example 2:

7a^2 - 4b + 11

In this expression: 7a^2 indicates that the variable a is squared (multiplied by itself), 7 is the coefficient for a^2, 4 is the coefficient for b, and 11 is a constant.

Evaluating algebraic expressions

To evaluate an algebraic expression, replace the variables with the given numerical values and then perform the arithmetic operations.

Example:

Evaluate 2x + 3 for x = 5.

Substitute 5 for x:

2(5) + 3 = 10 + 3 = 13

The expression evaluates to 13.

Visualization of algebraic expressions

Having a visual representation can help you understand algebraic expressions better. Think of expressions as lines or collections of lines. Below are simplified abstract representations using lines and arrows.

+3 Variable (x)

This diagram roughly represents x + 3 where the length of the line associated with x can vary depending on the value of x.

Combining like terms

Combine like terms to simplify algebraic expressions. Like terms are terms in an expression whose variables are raised to the same power.

Example:

Simplify 3x + 2x - 4y + y + 7.

3x and 2x are like terms. 4y and y are like terms. (3x + 2x) - (4y - y) + 7 = 5x - 3y + 7

Operations on algebraic expressions

You can perform many operations on algebraic expressions to simplify or manipulate them, including addition, subtraction, multiplication, and division.

Addition and subtraction

Add or subtract expressions by combining like terms. For example:

5x + 3 - (2x + 4)

= 5x + 3 - 2x - 4 = (5x - 2x) + (3 - 4) = 3x - 1

Multiplication

To multiply expressions, use the distributive property or the FOIL method (for binomials).

For example, multiply (2x + 3)(x + 5):

FOIL Method: = First: 2x * x = 2x^2 Outer: 2x * 5 = 10x Inner: 3 * x = 3x Last: 3 * 5 = 15 Combine: 2x^2 + 10x + 3x + 15 = 2x^2 + 13x + 15

Division

Dividing an algebraic expression usually involves distributing the division through the terms of the expression:

Divide 4x^2 + 8x by 2x:

= (4x^2 / 2x) + (8x / 2x) = 2x + 4

Practical applications

Algebraic expressions are used in a variety of real-life applications. They can represent relationships between quantities in science and finance, or provide a way to calculate values based on known quantities.

Examples from daily life:

An expression can represent the calculation in a shopping problem:

C = 2x + 3y where x is the number of items A bought at $2 per item and y at $3 per item. By inserting the values of x and y, you can calculate the total cost.

Conclusion

Algebraic expressions form the foundation of algebra and are essential in expressing mathematical ideas concisely and precisely. By understanding their components and how to manipulate them, you develop the skills necessary to solve more complex equations and real-world problems. With practice, you will become more comfortable and competent in working with algebraic expressions.


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Algebraic Expressions

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