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- https://math.libretexts.org/Courses/Nova_Scotia_Community_College/MATH_1043/02%3A_Algebra/2.02%3A_The_Fundamentals_of_Algebra/2.2.04%3A_Simplifying_Algebraic_ExpressionsThe commutative property allows us to change the order of multiplication without affecting the product or answer. The associative property allows us to regroup without affecting the product or answer.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_372%3A_College_Algebra_for_Calculus/09%3A_Systems_of_Equations_and_Inequalities/9.01%3A_Systems_of_Linear_Equations_-_Gaussian_EliminationThis section introduces systems of linear equations and explains how to solve them using Gaussian elimination. It covers the process of converting a system into row-echelon form through row operations...This section introduces systems of linear equations and explains how to solve them using Gaussian elimination. It covers the process of converting a system into row-echelon form through row operations and back-substitution to find the solution. Examples demonstrate solving systems with unique, infinite, or no solutions, providing a foundational technique for linear algebra and applications.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/02%3A_Matrices/2.02%3A_The_Inverse_of_a_MatrixThis page explores matrix operations, focusing on the identity matrix and matrix inverses, including their existence, uniqueness, and the method for finding inverses through augmented matrices and row...This page explores matrix operations, focusing on the identity matrix and matrix inverses, including their existence, uniqueness, and the method for finding inverses through augmented matrices and row operations. It provides examples illustrating both the derivation of inverses and scenarios where matrices lack inverses.
- https://math.libretexts.org/Courses/Western_Technical_College/PrePALS_PreAlgebra/04%3A_Introduction_to_Algebra/4.03%3A_Simplifying_Algebraic_ExpressionsThe commutative property allows us to change the order of multiplication without affecting the product or answer. The associative property allows us to regroup without affecting the product or answer.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_370%3A_Precalculus/09%3A_Systems_of_Equations_and_Inequalities/9.01%3A_Systems_of_Linear_Equations_-_Gaussian_EliminationUp until now, when we concerned ourselves with solving different types of equations there was only one equation to solve at a time. Given an equation ()=() f ( x ) = g ( x ) , we could check our solut...Up until now, when we concerned ourselves with solving different types of equations there was only one equation to solve at a time. Given an equation ()=() f ( x ) = g ( x ) , we could check our solutions geometrically by finding where the graphs of =() y = f ( x ) and =() y = g ( x ) intersect. The x -coordinates of these intersection points correspond to the solutions to the equation ()=() f ( x ) = g ( x ) , and the y -coordinates were largely ignored.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/07%3A_Coordinate_Trigonometry/7.03%3A_Introduction_to_Trigonometric_IdentitiesThis section introduces trigonometric identities, including definitions, examples, and practical applications. It covers how to determine if an equation is an identity and introduces the Ratio, Recipr...This section introduces trigonometric identities, including definitions, examples, and practical applications. It covers how to determine if an equation is an identity and introduces the Ratio, Reciprocal, and Pythagorean Identities, explaining their significance and application in Trigonometry. Through examples and checkpoints, it enhances understanding of these fundamental concepts, preparing readers for more advanced topics in Trigonometry.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/02%3A_Matrices/2.03%3A_Elementary_MatricesThis page covers the concept of elementary matrices, which are derived from the identity matrix using row operations. It details how these matrices are key in finding the inverse of matrices and expre...This page covers the concept of elementary matrices, which are derived from the identity matrix using row operations. It details how these matrices are key in finding the inverse of matrices and expresses a matrix as a product of elementary matrices. Properties of invertible matrices are discussed, including the conditions that an \(n \times n\) matrix must meet to be invertible, emphasizing the significance of row operations.
- https://math.libretexts.org/Courses/Western_Technical_College/PrePALS_Math_with_Business_Apps/04%3A_Introduction_to_Algebra/4.03%3A_Simplifying_Algebraic_ExpressionsThe commutative property allows us to change the order of multiplication without affecting the product or answer. The associative property allows us to regroup without affecting the product or answer.
- https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Stitz-Zeager)/08%3A_Systems_of_Equations_and_Matrices/8.01%3A_Systems_of_Linear_Equations-_Gaussian_EliminationUp until now, when we concerned ourselves with solving different types of equations there was only one equation to solve at a time. Given an equation ()=() f ( x ) = g ( x ) , we could check our solut...Up until now, when we concerned ourselves with solving different types of equations there was only one equation to solve at a time. Given an equation ()=() f ( x ) = g ( x ) , we could check our solutions geometrically by finding where the graphs of =() y = f ( x ) and =() y = g ( x ) intersect. The x -coordinates of these intersection points correspond to the solutions to the equation ()=() f ( x ) = g ( x ) , and the y -coordinates were largely ignored.
- https://math.libretexts.org/Bookshelves/PreAlgebra/Prealgebra_(Arnold)/03%3A_The_Fundamentals_of_Algebra/3.03%3A_Simplifying_Algebraic_ExpressionsThe commutative property allows us to change the order of multiplication without affecting the product or answer. The associative property allows us to regroup without affecting the product or answer.
- https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/04%3A_Geometry/4.01%3A_The_BasicsThe reader who has seen group theory will know that in addition to the three properties listed in our definition, the group operation must satisfy a property called associativity. In the context of tr...The reader who has seen group theory will know that in addition to the three properties listed in our definition, the group operation must satisfy a property called associativity. In the context of transformations, the group operation is composition of transformations, and this operation is always associative. So, in the present context of transformations, we omit associativity as a property that needs checking.
