Final answer:
The concepts mentioned relate to standard algebraic procedures for solving equations, which involve distributing constants, using properties of equality, and correctly applying multiplication and division rules. It is essential to simplify and check solutions for reasonableness throughout the process.
Step-by-step explanation:
The principles outlined in the question fall under the category of algebraic manipulation and simplification techniques, which are crucial in solving equations. The steps mentioned, such as distributing -1, utilizing the addition property of equality, and using the division property of equality, are all standard procedures in solving algebraic equations.
To solve an equation efficiently, one should first distribute any constants across terms within brackets to simplify the expression. Then, using the addition property of equality, terms can be moved from one side of the equation to the other to isolate the variable. Once the variable terms are isolated, the division property of equality is often applied to solve for the variable by dividing both sides of the equation by the coefficient of the variable. It is also vital to eliminate terms wherever possible and always check the solution to ensure it is reasonable.
While simplifying, remember that multiplication and division rules need to be applied correctly depending on the signs of the numbers involved. For instance, multiplying or dividing two positive numbers, or two negative numbers, results in a positive sign, while operations with numbers having opposite signs result in a negative sign. Also, if the numerator and denominator in a fraction are equal, that fraction simplifies to one.