Äquivalenzumformungen

Äquivalenzumformungen refers to a set of operations performed on an equation or inequality that preserve its solution set. In essence, these are transformations that ensure the resulting equation or inequality has precisely the same solutions as the original one. This concept is fundamental in algebra for solving equations and inequalities.

Common äquivalenzumformungen include adding or subtracting the same value from both sides of an equation or inequality. Multiplying or dividing both sides by the same non-zero value also maintains equivalence. However, care must be taken when multiplying or dividing by a variable expression, as this can sometimes introduce extraneous solutions or lose existing ones if the expression can be zero. When dealing with inequalities, multiplying or dividing by a negative number reverses the inequality sign.

The goal of applying äquivalenzumformungen is to isolate the variable of interest on one side of the equation or inequality, thereby revealing its value or the range of values it can take. Each step must be reversible to guarantee that the solution set remains unchanged. For instance, if you add 5 to both sides, you can subtract 5 from both sides of the new equation to return to the original. This reversibility is the hallmark of an äquivalenzumformung.