Page 2: Equation Identification and Structure

This page focuses on identifying equations and understanding their structure through practical examples and exercises.

Example: The equation x² = 25 has two solutions: x = +5 and x = -5

Definition: Solving an equation means determining all its possible solutions.

Highlight: The page includes exercises for identifying equations and breaking them down into their components (first member, second member, and unknown).

Page 3: Principles of Equivalence

This page explains the fundamental principles that govern equivalent equations and how to maintain equation equality.

Definition: Two equations are equivalent when they have exactly the same solutions.

Highlight: First Principle: Adding or subtracting the same expression to both sides of an equation results in an equivalent equation.

Example: 6x + 1 = 2x + 9 is equivalent to 6x + 1 + 3 = 2x + 9 + 3

Page 4: Rules for Solving Equations

This page outlines the essential rules used in solving equations, including transportation and cancellation rules.

Vocabulary: Transportation Rule (Regola del trasporto) - Terms can be moved from one side to another by changing their sign.

Example: In 3x = 12, the solution is found by dividing both sides by 3, giving x = 4

Highlight: The cancellation rule allows like terms to be combined on the same side of the equation.

Page 5: Understanding Equation Degrees

This page explains the concept of equation degrees and their classification.

Definition: The degree of an equation is determined by the highest power of the unknown variable.

Example: 5x² + 2 = 7x is a second-degree equation because the highest power of x is 2.

Highlight: First-degree equations have a maximum power of 1 for the unknown variable.

Page 6: Types of Solutions in First-Degree Equations

This page discusses the three possible types of solutions for first-degree equations.

Definition:

• Determined equation: Has exactly one solution
• Indeterminate equation: Has infinitely many solutions
• Impossible equation: Has no solutions

Example: For ax = b:

• If a ≠ 0, x = b/a (determined)
• If a = 0 and b = 0 (indeterminate)
• If a = 0 and b ≠ 0 (impossible)

Page 7: Solving Complex Equations

This page demonstrates solving more complex equations through step-by-step examples.

Example: 12x - 3 + 14 = -8x + 5 - 6x is solved by:

1. Combining like terms
2. Isolating variables
3. Solving for x

Highlight: The importance of systematic problem-solving and verification of solutions.

Page 8: Equivalent Equations

This page concludes with a deeper understanding of equivalent equations.

Definition: Equivalent equations are those that have exactly the same solutions.

Example: 3x = 6 is equivalent to x = 2, as both equations have the same solution.

Highlight: Understanding equivalent equations is crucial for solving complex mathematical problems.

Page 1: Introduction to Equations

This page introduces the fundamental concept of equations and their components. An equation is defined as an equality between two expressions, with at least one containing a variable (called the unknown).

Definition: An equation is an equality between two expressions where at least one contains a variable.

Vocabulary: The unknown (incognita) is the variable (x, y...) present in the expressions.

Example: In the equation 3x + x = 7, we have:

• First member: 3x + x
• Second member: 7
• Unknown: x

Highlight: A number is a solution (or root) of an equation if, when substituted for the unknown, it makes both sides of the equation equal.