Linear Equation Systems: Methods and Examples
Linear equation systems are a crucial topic in mathematics, particularly for students learning algebra. This page provides a comprehensive overview of various methods to solve these systems, including both algebraic and graphical approaches.
Definition: A linear equation system consists of two or more linear equations with at least two variables.
The page introduces three main algebraic methods for solving linear equation systems:
- Einsetzungsverfahren (Substitution Method)
- Gleichsetzungsverfahren (Equating Method)
- Additionsverfahren (Addition Method)
Example: For the system of equations:
I: 6x + 12y = 30
II: 3x + 3y = 9
Each method is demonstrated step-by-step to find the solution.
The substitution method involves rearranging one equation to express one variable in terms of the other, then substituting this into the second equation.
The equating method requires rearranging both equations to isolate the same variable, then equating these expressions.
The addition method involves adding or subtracting the equations to eliminate one variable.
Highlight: All three methods lead to the same solution: x = 1, y = 2.
The page also introduces the grafisches Lösungsverfahren (graphical solution method), which involves plotting the equations on a coordinate system and finding their intersection point.
Vocabulary:
- Sonderfälle (Special cases): Systems with no solution or infinitely many solutions.
- Unendlich viele Lösungen (Infinitely many solutions): When the equations represent the same line.
- Keine Lösung (No solution): When the equations represent parallel lines.
The document concludes with practice problems for each method, encouraging students to apply their understanding to various linear equation systems.
Example: A graphical solution problem is presented:
I: 2y - 2 = 6x
II: y + 2x = 6
Students are asked to solve this system graphically.
This comprehensive overview provides students with a solid foundation in solving linear equation systems, equipping them with multiple strategies to approach these mathematical problems.
