91Ó°ÊÓ

The distributive property is a fundamental concept in algebra that helps to simplify expressions by distributing multiplication over addition or subtraction. It states that for any numbers or variables, \(a(b+c) = ab + ac\). This rule allows us to break down and simplify complex expressions.

In our exercise, we start with the equation \(3(y-5) - 2y = -7\). Here, the distributive property is used to eliminate the parentheses by multiplying 3 with both \(y\) and \(-5\). So, \(3(y-5)\) becomes \(3y - 15\). It's a key first step in working towards solving the equation.

Applying the distributive property correctly ensures we rewrite the expression accurately, without changing its value. Be sure to distribute to every term inside the parentheses. This property is a vital tool that makes the rest of the procedure manageable.

After using the distributive property, the next step is to simplify the resulting expression by combining like terms. In algebra, 'like terms' are terms that have the same variable raised to the same power. For instance, in the expression \(3y - 15 - 2y\), \(3y\) and \(-2y\) are like terms because they both contain the variable \(y\) with the same exponent.

To combine like terms, you add or subtract their coefficients (the numerical part). So, combining \(3y\) and \(-2y\) in our example gives us \(y\), because \(3 - 2 = 1\). Simplifying in steps, we get from \(3y - 2y - 15 = -7\) to \(y - 15 = -7\).

This process reduces the equation to a simpler form, making it easier to solve. It's crucial to correctly identify and combine all like terms to avoid errors and find the correct solution efficiently.

The final step in solving a linear equation involves isolating the variable. This means getting the variable by itself on one side of the equation, which will give us the solution.

In our simplified equation \(y - 15 = -7\), we need to isolate \(y\). To do this, we perform operations that 'undo' the work of previous steps. Here, we add 15 to both sides of the equation because \(-15\) is subtracting from \(y\). So we have:

\(y - 15 + 15 = -7 + 15 \)
This simplifies to:
\(y = 8 \)
By adding the same number to both sides, we maintain the balance of the equation and correctly isolate \(y\).

Always remember, whatever you do to one side of the equation, do the same to the other. This keeps the equation balanced and leads you to the right answer. Isolating the variable is the step where you find the value that satisfies the initial equation.