Page 1: Solutions for Coordinate Geometry and Linear Equations

This page presents solutions for exercises from pages 266 and 239 of the Lambacher Schweizer Qualifikationsphase PDF. It covers coordinate geometry and linear equations.

The first problem involves finding coordinates of points P, Q, and R. The solution demonstrates the calculation process for determining these points in a three-dimensional coordinate system.

Example: P(101-613), Q(01013), R(51-75175)

The second part of the problem deals with solving a system of linear equations. The solution shows the step-by-step process of using substitution and elimination methods to find the values of variables.

Highlight: The use of a graphing calculator (GTR) is mentioned for solving linear systems, showcasing the integration of technology in advanced mathematics.

The page also includes a solution for problem 17a from page 239, which involves solving an equation with a parameter 'a'.

Vocabulary: Parameter - A variable whose value determines the specific form of a mathematical expression or equation.

The solution demonstrates how to isolate the parameter and solve for its value, resulting in a = -3.

Page 2: Vector Geometry and Parametric Equations

This page focuses on solutions from page 238, specifically problem 3b. It deals with vector geometry and parametric equations of lines.

The problem involves points A(-11210) and B(-3111) in three-dimensional space. The solution demonstrates how to find the direction vector of the line passing through these points.

Definition: Direction vector - A vector that indicates the direction of a line in space.

The page shows the process of setting up parametric equations for the line:

E: (-1) + t(-2) (2) (1) (-1) (-2)

This representation allows for the calculation of any point on the line using the parameter t.

Highlight: The solution also includes the process of finding the intersection of this line with a plane, demonstrating the integration of vector geometry and linear algebra concepts.

The final part of the solution involves solving a system of equations to find the coordinates of the intersection point:

4x + 8y = -4 x + 2z = -1

This showcases the application of Lambacher Schweizer Qualifikationsphase Lösungen in solving complex geometric problems using algebraic methods.