Final answer:
Algebra involves the study of variables, expressions, equations, and functions, and includes properties like commutative, associative, and distributive which allow manipulation of expressions. Algebra is applied across various fields, including finance, trigonometry, geometry, and statistics, and is fundamental in sciences where it is used to solve equations with multiple variables.
Step-by-step explanation:
Algebra is a branch of mathematics that deals with variables, algebraic expressions, equations, functions, and their properties. Algebraic expressions are combinations of variables, numbers, and operation symbols. The study of algebra allows us to understand relationships between quantities and solve equations where certain quantities are unknown.
Key properties of algebra include the commutative, associative, and distributive properties. These foundational properties allow for the manipulation and simplification of expressions. For instance, the commutative property states that a + b = b + a and ab = ba, meaning that the order of addition or multiplication does not change the result. The associative property allows us to change grouping in expressions such as (a + b) + c = a + (b + c) and (ab)c = a(bc). The distributive property connects multiplication and addition through a(b + c) = ab + ac.
Algebra also encompasses the concept of exponents, number patterns, and the analysis of functions. It applies to numerous fields, including finance and growth, trigonometry, analytical geometry, and statistics. Algebraic techniques are essential in sciences like chemistry and physics, where equations often include multiple variables and units. Mastery of algebra is crucial for managing equations to solve for unknowns and for understanding changes in systems, be it in thermodynamics or vector addition.