Understanding Basic Equations and Linear Systems

This introductory section establishes fundamental concepts of equations and their solutions. The content focuses on the definition of equations and various solving strategies.

Definition: An equation is a mathematical expression where two terms are connected by an equals sign (T₁ = T₂).

Example: Basic equations include x² = 4 and 7x + 3 = 1.

Highlight: Solutions are all values of variables that make the equation true.

Vocabulary: Linear equations can be expressed in the form ax + b = 0, where:

• ax is the linear term
• b is the constant term
• a,b ∈ R; a ≠ 0

Solution Methods for Linear Systems

The second page introduces systematic approaches to solving systems of linear equations, particularly focusing on practical applications.

Example: A problem involving 183 representatives with specific conditions:

• OVP has 13 fewer mandates than SPÖ
• OVP has 8 times as many mandates as FPÖ

Highlight: Three main solution methods are presented:

1. Substitution method
2. Addition method (elimination)
3. Comparison method

Definition: A linear system of equations with two variables consists of: ax + by = c dx + ey = f where a,b,c,d,e,f ∈ R and a,b,d,e ≠ 0