Lösungen

Lösungen is the plural form of die Lösung in German. The word has multiple related meanings, including an answer to a problem, and a chemical mixture formed when a solute dissolves in a solvent. The term derives from lösen, meaning to loosen or release.

In chemistry, a Lösung (the German term for solution) is a homogeneous mixture formed when a solute dissolves in a solvent. The composition is uniform at the molecular level. Key terms include solute, solvent, and concentration, with common units such as molarity, molality, and mass percent. Solubility describes how much solute can dissolve in a given amount of solvent at a given temperature, and can depend on temperature and, for gases, pressure. Properties of solutions include colligative effects like boiling-point elevation. Common examples are salt in water and sugar in water. Solutions can be liquids, but gases and solids can also form solutions, such as air or brass.

In mathematics, a Lösung refers to an assignment of values that satisfies an equation or system of equations. The collection of all solutions is called the Lösungsmenge (solution set). Common methods to find solutions include substitution and elimination for linear systems, factoring and completing the square for equations, and numerical methods for more complex problems. Existence and uniqueness theorems describe when solutions exist and whether they are unique. Example: the solution of x + 3 = 7 is x = 4.

In everyday language and in applied fields, Lösungen denote answers, proposed methods, or remedies to problems. In education, exercises require presenting a solution. In business and technology, the term Lösungs is used in compounds such as Lösungsvorschläge (proposed solutions) and Systemlösungen (system solutions).