Linear Equations with Two Variables

This section introduces the concept of jedno równanie z dwiema niewiadomymi (linear equations with two variables) and their graphical representations.

Key points:

• A linear equation with two variables is represented as ax + by = c, where a and b are not both zero
• The graph of a linear equation is a straight line
• Examples are provided, such as 3x + 2y = 4 and 2x - y = 5

Definition: A linear equation with two variables is an equation that can be written in the form ax + by = c, where a, b, and c are constants, and a and b are not both zero.

Example: The equation 2x - y = 5 can be rearranged to y = 2x - 5, which is the slope-intercept form of a line.

The section also introduces układy równań liniowych z dwiema niewiadomymi zadania (systems of linear equations with two variables) and their graphical solutions.

Highlight: The solution to a system of equations is a pair of numbers (x, y) that satisfies both equations simultaneously.

Solving Systems of Equations

This part covers various methods for rozwiązywanie układów równań pierwszego stopnia z dwiema niewiadomymi (solving first-degree systems of equations with two variables).

Methods discussed include:

1. Rozwiązywanie układów równań metodą podstawiania (Substitution method)
2. Rozwiązywanie układów równań metodą przeciwnych współczynników (Elimination method)
3. Graficzne rozwiązywanie układów równań (Graphical method)

Example: Using the substitution method to solve the system: 2x - 3y = 1 3x - 4y = 11

The section provides step-by-step solutions for various types of systems, including:

• Systems with one unique solution (consistent independent)
• Systems with no solution (inconsistent)
• Systems with infinitely many solutions (consistent dependent)

Vocabulary:

• Układ oznaczony: A system with one unique solution
• Układ sprzeczny: A system with no solution
• Układ nieoznaczony: A system with infinitely many solutions