91Ó°ÊÓ

Understanding how to combine like terms is essential for simplifying algebraic equations. Like terms are terms that contain the same variables raised to the same power. By combining them, you consolidate the equation, making it easier to solve.

Let's take the equation from our exercise, where we have two terms on the right side, both with the variable x: cx and -dx. Since they have the same variable, we can combine them by adding or subtracting their coefficients. In this case, we would perform c - d to get a single term with x.

Why is this helpful? By combining like terms, you reduce the complexity of the equation and clarify what you're solving for. In practical terms, think of it like combining coins; instead of dealing with a pile of nickels and dimes separately, if you combine them, you get a clear sum to work with, which is much simpler to use.

The distributive property is a tool that lets you multiply a single term and a group of terms inside a set of parentheses. Imagine you have an equation like a(b + c). Using the distributive property, you'd multiply a by b and a by c to get ab + ac.

In our exercise, we saw the distributive property in reverse. We started with cx - dx, which looks like it has been distributed with x being the common factor. Recognizing this, we reverse the process to simplify the equation to (c - d)x. This reverse distribution combines the like terms and prepares us to solve for x by isolating the variable. Remember, the distributive property works both ways and is a critical step in the simplification process.

To isolate the variable means to get the variable you are solving for by itself on one side of the equation. The goal is to have the variable, such as x, with a coefficient of 1.

In our exercise, after combining like terms using the distributive property, we ended with the equation y = (c - d)x. To isolate x, we need to get rid of the (c - d) that's multiplied by x. We do this by dividing both sides of the equation by (c - d). What we are doing here is performing the opposite operation to both sides of the equation to maintain the balance. When we divide y by (c - d), we are left with x = y/(c - d).

Isolating the variable is like finding out the single type of fruit in a bag that equally distributes into the number of available baskets. You're finding out how much of that one thing fits into each space to make the equation balanced. By doing so, x stands alone, and we have our solution.