91Ó°ÊÓ

At the heart of algebra lie algebraic equations, which are mathematical statements indicating that two expressions are equal. For instance, take the equation from our exercise:
\[ 10(-4+y) = 2y \].
In this case, each side of the equal sign represents an expression, and our job is to find the value of the variable 'y' that makes the equation true. In algebraic equations, variables can appear on one or both sides of the equation, and understanding how to manipulate these equations is key to finding the solution. To tackle these equations, we rely on a variety of principles and properties, such as the distributive property and the methods used to isolate variables which will be discussed further.

A fundamental tool in our algebraic toolkit is the distributive property, which is used to remove parentheses from an expression. It states that you can multiply a single term by each term inside a set of parentheses, which was our initial step in the exercise.
For example:
\[ a(b + c) = ab + ac \].
The distributive property is essential when you need to simplify algebraic expressions before you solve for the variable. In our exercise, we applied this property as follows:
\[ 10(-4+y) = 2y \] becomes \[ 10(-4) + 10y = 2y \],
which simplifies to \[ -40 + 10y = 2y \].
Understanding and correctly applying the distributive property allows us to proceed to the next step in solving equations: collecting like terms and isolating the variable.

Isolating the variable is essentially the goal in solving an algebraic equation. This involves moving all terms containing the variable to one side and all other terms to the opposite side. The purpose is to have the variable alone on one side of the equation, thereby 'isolating' it to find its value. This process typically involves addition or subtraction to remove terms from one side, followed by multiplication or division to get the variable by itself.
In the provided step-by-step solution, we first subtracted \(2y\) from both sides to collect like terms:
\[ -40 + 10y - 2y = 0 \],
which simplifies to:
\[ 8y = -40 \].
Then, to isolate 'y', we divided both sides by the coefficient of 'y', which is 8:
\[ y = \frac{-40}{8} \].
Through division, we've successfully isolated the variable and determined its value. Mastery of isolating the variable makes algebra much more manageable and is crucial for solving more complex equations.