Solving For W: 6w + 5(w + 6) = -3

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Hey guys! Today, we're diving into a classic algebra problem: solving for the variable w in the equation 6w + 5(w + 6) = -3. Don't worry, it's not as intimidating as it looks! We'll break it down step-by-step, so you can follow along and master these types of problems. Understanding how to solve for variables is a fundamental skill in mathematics, and it opens the door to more complex equations and concepts. So, let's get started and unlock the mystery of w!

Understanding the Equation

Before we jump into solving, let's take a closer look at the equation itself: 6w + 5(w + 6) = -3. The core of algebra lies in manipulating equations to isolate the variable we're interested in – in this case, w. Think of it like peeling away layers of an onion; we need to undo the operations performed on w until we have it all by itself on one side of the equation. This involves using the properties of equality, which basically say that whatever you do to one side of the equation, you must also do to the other side to keep things balanced. We'll be using these properties throughout our solution. The left side of the equation contains terms with w and a set of parentheses, which indicates we'll need to use the distributive property. The right side is simply a constant, -3. Our goal is to simplify the left side, combine like terms, and then isolate w. Remember, each step we take brings us closer to our final answer, so let's move on to the first step: simplifying the equation.

Step 1: Distribute

The first key step in solving this equation is to tackle those parentheses. We need to apply the distributive property, which means multiplying the 5 outside the parentheses by each term inside. So, 5(w + 6) becomes 5 * w + 5 * 6, which simplifies to 5w + 30. Now, let's rewrite the entire equation with this simplification: 6w + 5w + 30 = -3. See? We've already made progress! By distributing, we've eliminated the parentheses and made the equation easier to work with. This is a crucial technique in algebra, as it allows us to combine terms and simplify expressions. Remember, the distributive property is like sharing – the number outside the parentheses gets multiplied by each term inside. Next up, we'll combine the like terms on the left side of the equation to further simplify things. Stay with me, we're on the right track!

Step 2: Combine Like Terms

Now that we've distributed and gotten rid of the parentheses, it's time to combine like terms. Look at the left side of our equation: 6w + 5w + 30 = -3. We have two terms that involve the variable w: 6w and 5w. These are like terms, meaning they have the same variable raised to the same power (in this case, w to the power of 1). To combine them, we simply add their coefficients: 6 + 5 = 11. So, 6w + 5w becomes 11w. Our equation now looks like this: 11w + 30 = -3. Isn't it getting simpler? Combining like terms helps us streamline the equation and makes it easier to isolate our variable, w. It's like tidying up before we get to the main event. Now that we've combined like terms, we're one step closer to isolating w. Let's move on to the next step: isolating the variable term.

Step 3: Isolate the Variable Term

Our goal is to get w all by itself on one side of the equation. Currently, we have 11w + 30 = -3. The term with w is 11w, and we need to isolate it. To do this, we need to get rid of the +30. Remember the properties of equality? What we do to one side, we must do to the other. So, to get rid of the +30, we subtract 30 from both sides of the equation: 11w + 30 - 30 = -3 - 30. This simplifies to 11w = -33. Awesome! We've successfully isolated the variable term, 11w. We're getting closer and closer to finding the value of w. Isolating the variable term is a crucial step because it sets us up to perform the final operation: dividing to solve for w. Are you ready for the final step? Let's do it!

Step 4: Solve for w

We're in the home stretch! We have 11w = -33. To finally solve for w, we need to get rid of the 11 that's multiplying it. Again, we use the properties of equality. To undo multiplication, we divide. So, we divide both sides of the equation by 11: (11w) / 11 = -33 / 11. This simplifies to w = -3. Boom! We've done it! We've successfully solved for w. The value of w that makes the equation 6w + 5(w + 6) = -3 true is -3. Solving for a variable involves a series of steps, each building upon the previous one. Remember, the key is to isolate the variable by using inverse operations and the properties of equality. Now that we've found our solution, it's always a good idea to check our work.

Step 5: Check Your Solution

It's always a good idea to double-check your work, especially in math! To verify our solution, we'll substitute w = -3 back into the original equation: 6w + 5(w + 6) = -3. Let's plug in -3 for w: 6(-3) + 5(-3 + 6) = -3. Now, we simplify: -18 + 5(3) = -3. Continue simplifying: -18 + 15 = -3. Finally, we get -3 = -3. This is a true statement! This confirms that our solution, w = -3, is correct. Checking your solution is a fantastic habit to develop because it helps you catch any errors and build confidence in your answers. It's like having a built-in safety net! Plus, it reinforces your understanding of the steps involved in solving the equation. So, never skip this step if you have the time.

Conclusion

Alright guys, we did it! We successfully solved the equation 6w + 5(w + 6) = -3 and found that w = -3. We walked through each step, from distributing and combining like terms to isolating the variable and checking our answer. Remember, solving algebraic equations is all about using the properties of equality to manipulate the equation and isolate the variable. This problem showcases several key algebraic techniques, including the distributive property, combining like terms, and using inverse operations. These are fundamental skills that you'll use again and again in your math journey. Practice makes perfect, so try solving similar equations to solidify your understanding. And if you ever get stuck, don't hesitate to break the problem down into smaller steps, just like we did today. Keep up the great work, and I'll see you in the next math adventure!