Solving a System of Linear Equations with Three Variables

This page demonstrates the process of solving a system of linear equations with three variables using the triangle system method. The problem involves finding the coefficients of a quadratic function f$x$ = ax² + bx + c, given three equations.

The solution process is as follows:

1. The initial system of equations is presented:

   • a + b + c = 15
   • 3a - b = 15
   • 12a = 48

2. The triangle system method is applied to solve for the variables a, b, and c.

3. The solution steps are shown in detail, including algebraic manipulations and substitutions.

4. The final values for a, b, and c are determined:

   • a = 4
   • b = -27
   • c = -16

5. The resulting quadratic function is presented: f$x$ = 4x² - 27x - 16

Vocabulary: Triangle system (Dreieckssystem) - A method for solving systems of linear equations by transforming them into a triangular form.

Example: The system of equations is solved step-by-step, demonstrating how to use the triangle system method to find the values of a, b, and c.

Highlight: The final quadratic function f$x$ = 4x² - 27x - 16 is obtained by substituting the solved values of a, b, and c into the general form ax² + bx + c.

Definition: A system of linear equations with three variables is a set of equations involving three unknown quantities, typically represented by letters such as x, y, and z or, in this case, a, b, and c.

This detailed solution provides a clear example of how to solve Gleichungssysteme mit 3 Variablen (systems of equations with 3 variables) and can be useful for students learning about lineare Gleichungssysteme lösen (solving linear systems of equations).