äquivalenzumformung

Äquivalenzumformung, also known as equivalent transformation or algebraic manipulation, is a fundamental concept in mathematics. It refers to the process of rewriting an equation or inequality into a simpler or more convenient form while preserving its original solutions. This means that the transformed equation or inequality has exactly the same set of solutions as the original one.

The core principle behind äquivalenzumformung is to perform operations on both sides of an equation or inequality that do not alter the truth set. Common operations include adding or subtracting the same value to both sides, multiplying or dividing both sides by the same non-zero value, and substituting equivalent expressions. For example, in the equation x + 2 = 5, subtracting 2 from both sides results in x = 3. Both x + 2 = 5 and x = 3 have the same solution, which is x = 3.

Äquivalenzumformung is crucial for solving equations and inequalities, simplifying expressions, and proving mathematical statements. It forms the basis for many algebraic techniques, allowing mathematicians to systematically manipulate mathematical expressions to isolate variables, identify relationships, and arrive at desired conclusions. Understanding and applying äquivalenzumformung correctly is essential for success in algebra and higher-level mathematics.