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Math for Grade 10
1 Number Systems
1-1 Introduction to Number Systems
1-2 Types of Numbers
1-2 1 Natural Numbers
1-2 2 Whole Numbers
1-2 3 Integers
1-2 4 Rational Numbers
1-2 5 Irrational Numbers
1-2 6 Real Numbers
1-3 Properties of Numbers
1-3 1 Commutative Property
1-3 2 Associative Property
1-3 3 Distributive Property
1-3 4 Identity Property
1-3 5 Inverse Property
1-4 Operations with Real Numbers
1-4 1 Addition
1-4 2 Subtraction
1-4 3 Multiplication
1-4 4 Division
1-4 5 Order of Operations (PEMDASBODMAS)
1-5 Exponents and Radicals
1-5 1 Exponent Rules
1-5 2 Scientific Notation
1-5 3 Square Roots
1-5 4 Cube Roots
1-5 5 nth Roots
1-6 Rationalizing Denominators
2 Algebra
2-1 Introduction to Algebra
2-2 Expressions and Equations
2-2 1 Simplifying Algebraic Expressions
2-2 2 Linear Equations
2-2 3 Quadratic Equations
2-2 4 Solving Equations with Variables on Both Sides
2-2 5 Solving Literal Equations
2-3 Inequalities
2-3 1 Linear Inequalities
2-3 2 Quadratic Inequalities
2-3 3 Absolute Value Inequalities
2-4 Polynomials
2-4 1 Introduction to Polynomials
2-4 2 Adding and Subtracting Polynomials
2-4 3 Multiplying Polynomials
2-4 4 Factoring Polynomials
2-4 5 Special Products
2-5 Rational Expressions
2-5 1 Simplifying Rational Expressions
2-5 2 Multiplying and Dividing Rational Expressions
2-5 3 Adding and Subtracting Rational Expressions
2-5 4 Solving Rational Equations
2-6 Functions
2-6 1 Introduction to Functions
2-6 2 Function Notation
2-6 3 Graphing Functions
2-6 4 Linear Functions
2-6 5 Quadratic Functions
2-6 6 Polynomial Functions
2-6 7 Rational Functions
3 Geometry
3-1 Introduction to Geometry
3-2 Basic Geometric Figures
3-2 1 Points, Lines, and Planes
3-2 2 Angles
3-2 3 Triangles
3-2 4 Quadrilaterals
3-2 5 Circles
3-3 Geometric Properties and Relationships
3-3 1 Congruence and Similarity
3-3 2 Pythagorean Theorem
3-3 3 Triangle Inequality Theorem
3-4 Perimeter, Area, and Volume
3-4 1 Perimeter of Polygons
3-4 2 Area of Polygons
3-4 3 Area of Circles
3-4 4 Surface Area of Solids
3-4 5 Volume of Solids
3-5 Transformations
3-5 1 Translations
3-5 2 Reflections
3-5 3 Rotations
3-5 4 Dilations
4 Trigonometry
4-1 Introduction to Trigonometry
4-2 Trigonometric Ratios
4-2 1 Sine, Cosine, and Tangent
4-2 2 Reciprocal Trigonometric Functions
4-3 Solving Right Triangles
4-3 1 Using Trigonometric Ratios to Solve Right Triangles
4-3 2 Applications of Right Triangle Trigonometry
4-4 Trigonometric Identities
4-4 1 Pythagorean Identities
4-4 2 Angle Sum and Difference Identities
4-4 3 Double Angle Identities
4-5 Graphing Trigonometric Functions
4-5 1 Graphing Sine and Cosine Functions
4-5 2 Graphing Tangent Functions
4-5 3 Transformations of Trigonometric Graphs
5 Statistics and Probability
5-1 Introduction to Statistics
5-2 Data Collection and Representation
5-2 1 Types of Data
5-2 2 Frequency Distributions
5-2 3 Graphical Representations of Data
5-3 Measures of Central Tendency
5-3 1 Mean
5-3 2 Median
5-3 3 Mode
5-4 Measures of Dispersion
5-4 1 Range
5-4 2 Variance
5-4 3 Standard Deviation
5-5 Probability
5-5 1 Introduction to Probability
5-5 2 Basic Probability Concepts
5-5 3 Probability of Compound Events
5-5 4 Conditional Probability
5-6 Statistical Inference
5-6 1 Sampling and Sampling Distributions
5-6 2 Confidence Intervals
5-6 3 Hypothesis Testing
2-2 Expressions and Equations Explained

2-2 Expressions and Equations Explained

Key Concepts of Expressions and Equations

Expressions and equations are fundamental components in algebra. Understanding the difference between them and how to manipulate them is crucial for solving mathematical problems.

1. Expressions

An expression is a combination of numbers, variables, and mathematical operations (such as addition, subtraction, multiplication, and division) without an equal sign. Expressions can be simplified but not solved.

Example:

\[ 3x + 5 \]

This is an expression because it does not have an equal sign.

2. Equations

An equation is a statement that two expressions are equal. It contains an equal sign and can be solved to find the value of the variable.

Example:

\[ 3x + 5 = 14 \]

This is an equation because it has an equal sign and can be solved for \( x \).

3. Simplifying Expressions

Simplifying an expression involves performing all possible operations to reduce the expression to its simplest form. This often involves combining like terms and using the order of operations (PEMDAS/BODMAS).

Example:

Simplify \( 4x + 7 - 2x + 3 \):

Combine like terms: \( 4x - 2x + 7 + 3 = 2x + 10 \).

4. Solving Equations

Solving an equation involves finding the value of the variable that makes the equation true. This is done by isolating the variable on one side of the equation.

Example:

Solve \( 3x + 5 = 14 \):

Subtract 5 from both sides: \( 3x = 9 \).

Divide both sides by 3: \( x = 3 \).

5. Combining Like Terms

Combining like terms is a method used to simplify expressions by adding or subtracting terms that have the same variable and exponent.

Example:

Combine like terms in \( 5x^2 + 3x - 2x^2 + 4 \):

Combine \( 5x^2 \) and \( -2x^2 \): \( 3x^2 \).

The simplified expression is \( 3x^2 + 3x + 4 \).

6. Distributive Property

The distributive property is used to simplify expressions and solve equations by distributing a factor across a sum or difference.

Example:

Use the distributive property to simplify \( 3(2x + 4) \):

Distribute 3: \( 3 \times 2x + 3 \times 4 = 6x + 12 \).

Practical Applications

Understanding expressions and equations is essential for solving real-world problems. For instance, calculating the cost of items with discounts, determining the area of a rectangle, or solving for the unknown in a physics formula all involve expressions and equations.

Example:

If a store offers a 20% discount on a $50 item, you can set up an equation to find the final price:

Let \( x \) be the final price after the discount:

\[ x = 50 - 0.20 \times 50 \]

Simplify: \( x = 50 - 10 = 40 \).

The final price is $40.

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