91Ó°ÊÓ

The distributive property is a key algebraic concept used to simplify equations and eliminate parentheses. In this exercise, it appears in the first step where we distribute the 9 over the terms inside the parentheses. The original equation is:
\[ 9(y-2)-8y=-16 \]
Using the distributive property, we multiply 9 by both terms inside the parentheses: that means we do: 9 \times y and 9 \times -2. When done, the equation looks like this:
\[ 9y - 18 - 8y = -16 \]
This operation simplifies the equation and makes it easier to solve by removing the parentheses.
In general, the distributive property says that for any numbers a, b, and c:
\[ a(b + c) = ab + ac \]
It's a vital tool for solving multi-step equations.

After using the distributive property, the next step is to combine like terms. Like terms are terms that contain the same variable raised to the same power. In the equation, we have:\[ 9y - 18 - 8y = -16 \]
Here, 9y and -8y are like terms because they both contain the variable y. By combining them, we get:\[ y - 18 = -16 \]
This means we subtract 8y from 9y, resulting in y. Combining like terms simplifies the equation further, making it easier to isolate the variable in the next step. Ignore the other terms for now and focus on simplifying the variable terms.
Always look for like terms in all steps of solving equations as it helps in clearing up any extra terms and simplifying the equation.

The final essential step in solving the equation is isolating the variable, which means getting the variable by itself on one side of the equation. Here, we start from:\[ y - 18 = -16 \]
To isolate y, we need to remove -18 from the left side. We do this by adding 18 to both sides of the equation:
\[ y - 18 + 18 = -16 + 18 \]
This simplifies to:\[ y = 2 \]
Adding the same number to both sides ensures the equation remains balanced. Now, the variable y is isolated, and we have our solution.
In general, isolating the variable often involves performing inverse operations (like adding if there's subtraction or dividing if there's multiplication) to both sides of the equation to keep it balanced and solve for the unknown variable.