Solving Equations: A Step-by-Step Guide

Hey Plastik Magazine readers! Ever get stuck on a math problem that looks like it's written in a different language? Don't worry, we've all been there. Today, we're going to break down a common type of equation and show you exactly how to solve it. We'll use a specific example, the equation $2(3z+2)-7=51$, but the steps we use will work for tons of other equations too. So, grab your pencils, and let's dive into the world of equation solving!

Understanding the Equation: $2(3z+2)-7=51$

Okay, before we jump into solving, let's take a good look at what we're dealing with. The equation $2(3z+2)-7=51$ might seem a little intimidating at first, but it's really just a puzzle waiting to be solved. Our main goal here is to find the value of the variable 'z' that makes this equation true. Think of it like a balancing act: we want to figure out what 'z' needs to be so that both sides of the equals sign (=) weigh the same. Now, let’s break down each part of the equation to understand it better. First up, we have parentheses. The term inside the parentheses, $(3z + 2)$, is a group of terms that we need to handle carefully. The '3z' means '3 times z', and the '+ 2' means we're adding 2 to that result. It's super important to remember the order of operations here, which is often remembered by the acronym PEMDAS (or BODMAS). This tells us we need to deal with parentheses first. Next, we see that the entire expression inside the parentheses is being multiplied by 2. This is a key step because it means we'll need to use the distributive property soon. Then, we subtract 7 from the result of the multiplication. This subtraction is another operation we need to consider in our solving process. On the right side of the equation, we have the number 51. This is our target – the value that the left side of the equation needs to equal once we've figured out what 'z' is. In summary, we're trying to find the value of 'z' that, when plugged into the equation, will make the left side equal to 51. To do this effectively, we'll need to follow a step-by-step process, which we’ll get into next. So, stay with us, and we'll make solving this equation feel like a breeze!

Step 1: Distribute the 2

The first thing we need to tackle in our equation $2(3z+2)-7=51$ is those pesky parentheses. Remember, the order of operations (PEMDAS/BODMAS) tells us to deal with parentheses first. But we can't just magically make them disappear! Instead, we need to use the distributive property. What this property basically says is that if you have a number multiplied by a group inside parentheses, you need to multiply that number by each term inside the parentheses. So, in our case, we have 2 multiplied by $(3z + 2)$. This means we need to multiply 2 by 3z and then multiply 2 by 2. Let's break it down: 2 * 3z equals 6z. Think of it as having two groups of 3z, which gives you a total of 6z. Next, 2 * 2 equals 4. This is straightforward multiplication. Now, we can rewrite our equation by replacing $2(3z + 2)$ with $6z + 4$. Our equation now looks like this: $6z + 4 - 7 = 51$. See how we've gotten rid of the parentheses? That's a big step forward! We've essentially expanded the expression, making it easier to work with. This step is crucial because it simplifies the equation and allows us to start isolating the variable 'z'. Without distributing, we'd be stuck with the parentheses, and it would be much harder to move forward. Remember, the distributive property is your friend when you see parentheses like this. It helps you break down complex expressions into simpler ones. So, to recap, we took the 2 outside the parentheses and multiplied it by each term inside, resulting in $6z + 4$. This is a key move in solving the equation, and it sets us up for the next steps. Now that we've distributed, we can move on to combining like terms and further simplifying the equation. Keep following along, and we'll have this equation solved in no time!

Step 2: Combine Like Terms

Alright, we've successfully distributed the 2 in our equation, and we're one step closer to finding the value of 'z'. Our equation currently looks like this: $6z + 4 - 7 = 51$. The next thing we want to do is to combine like terms. What does that mean, exactly? Well, like terms are simply terms in the equation that can be combined because they are similar. In this case, we have two constant terms (numbers without a variable): +4 and -7. These are like terms because they're both just plain numbers. We can combine them by performing the indicated operation, which is subtraction in this case. So, let's do it! We have 4 - 7, which equals -3. Think of it like starting at 4 on a number line and then moving 7 spaces to the left – you'll end up at -3. Now, we can rewrite our equation again, replacing the $4 - 7$ with -3. Our equation now looks like this: $6z - 3 = 51$. See how much simpler it's becoming? By combining like terms, we've reduced the number of individual terms in the equation, making it easier to isolate 'z'. This is a crucial step because it streamlines the equation and brings us closer to our solution. The term '6z' remains unchanged because it doesn't have any other like terms to combine with. It's the term with the variable, and we're ultimately trying to get it by itself on one side of the equation. Combining like terms is a fundamental skill in algebra, and it's something you'll use all the time when solving equations. It's all about simplifying the equation to its most basic form before you start isolating the variable. So, to recap, we identified the like terms (+4 and -7), combined them to get -3, and rewrote our equation as $6z - 3 = 51$. We're making great progress! Now, we're ready to move on to the next step, which involves isolating the term with the variable. Let's keep going!

Step 3: Isolate the Variable Term

Okay, guys, we're making fantastic progress! We've distributed, combined like terms, and now our equation looks like this: $6z - 3 = 51$. The next key step is to isolate the variable term. This means we want to get the term with 'z' (in this case, 6z) all by itself on one side of the equation. Think of it like building a wall around '6z', separating it from everything else. Right now, we have a '- 3' on the same side as the '6z'. We need to get rid of that -3. How do we do it? We use the magic of inverse operations! Remember, whatever we do to one side of the equation, we must do to the other side to keep the equation balanced. Since we have a subtraction of 3, the inverse operation is addition. So, we're going to add 3 to both sides of the equation. This is a crucial move because it will cancel out the -3 on the left side, leaving us with just the variable term. Let's write it out: $6z - 3 + 3 = 51 + 3$. On the left side, the -3 and +3 cancel each other out, leaving us with just 6z. On the right side, 51 + 3 equals 54. So, our equation now looks like this: $6z = 54$. Wow! Look how much simpler it's become. We've successfully isolated the variable term, and we're just one step away from solving for 'z'. This step of isolating the variable term is super important in solving equations. It's all about strategically using inverse operations to undo any operations that are happening on the same side as the variable. By adding 3 to both sides, we effectively moved the -3 to the other side of the equation, but as a +3. This is a common technique in algebra, and it's something you'll use again and again. So, to recap, we added 3 to both sides of the equation to isolate the variable term, resulting in the equation $6z = 54$. We're in the home stretch now! Let's move on to the final step and find the value of 'z'. You've got this!

Step 4: Solve for the Variable

Alright, Plastik Magazine crew, we've reached the final stage! We've distributed, combined like terms, isolated the variable term, and now our equation is a sleek $6z = 54$. We are so close to cracking this equation! Remember, our ultimate goal is to solve for the variable, which means we want to find out what 'z' equals. Right now, we have 6 multiplied by z. To get 'z' all by itself, we need to undo that multiplication. And what's the inverse operation of multiplication? You guessed it – division! Just like in the previous step, whatever we do to one side of the equation, we must do to the other side to maintain balance. So, we're going to divide both sides of the equation by 6. Let's write it out: $(6z) / 6 = 54 / 6$. On the left side, the 6 in the numerator and the 6 in the denominator cancel each other out, leaving us with just 'z'. This is exactly what we wanted! On the right side, 54 divided by 6 equals 9. So, our equation now looks like this: $z = 9$. Boom! We've solved it! We've found the value of 'z' that makes the original equation true. This final step of dividing to isolate the variable is a classic technique in algebra. It's all about undoing the multiplication or division that's attached to the variable. By dividing both sides by the coefficient (the number in front of the variable), we effectively isolate the variable and reveal its value. This is the moment of truth, the culmination of all our hard work! So, to recap, we divided both sides of the equation by 6 to solve for 'z', resulting in the solution $z = 9$. We did it! We've successfully solved the equation $2(3z+2)-7=51$. But wait, there's one more thing we can do to be absolutely sure we've got the right answer.

Step 5: Check Your Solution

Okay, we've done all the hard work and found that $z = 9$ is the solution to our equation. But before we declare victory, there's one super important step we need to take: check our solution. This is like the quality control step of equation solving. It ensures that we haven't made any mistakes along the way and that our answer is actually correct. So, how do we check our solution? It's simple! We take the value we found for 'z' (which is 9) and plug it back into the original equation. This is crucial – we need to go back to the very beginning, to the equation $2(3z+2)-7=51$. Now, we'll replace every 'z' in the equation with 9: $2(3(9)+2)-7=51$. Now, we just need to simplify the left side of the equation, following the order of operations (PEMDAS/BODMAS). First, we deal with the parentheses. Inside the parentheses, we have $3(9) + 2$. 3 times 9 is 27, so we have $27 + 2$, which equals 29. Now our equation looks like this: $2(29) - 7 = 51$. Next, we perform the multiplication: 2 times 29 equals 58. So our equation becomes: $58 - 7 = 51$. Finally, we do the subtraction: 58 minus 7 equals 51. So, we have $51 = 51$. Hooray! The left side of the equation equals the right side of the equation. This means our solution, $z = 9$, is correct! If we had gotten a different number on the left side, it would mean we made a mistake somewhere, and we'd need to go back and check our steps. Checking your solution is like having a safety net. It gives you the confidence that you've solved the equation correctly. It's a habit that all good equation solvers develop. So, to recap, we plugged our solution, $z = 9$, back into the original equation and simplified. We found that both sides of the equation were equal, confirming that our solution is correct. Awesome job, everyone! We've successfully solved and checked our solution for this equation.

Conclusion: You're an Equation-Solving Pro!

And there you have it, Plastik Magazine math enthusiasts! We've taken a potentially intimidating equation, $2(3z+2)-7=51$, and broken it down into manageable steps. We've distributed, combined like terms, isolated the variable term, solved for the variable, and even checked our solution. You've now got a solid understanding of how to tackle this type of equation. Remember, the key to success in algebra (and in life!) is to break down complex problems into smaller, more manageable steps. Don't be afraid to take your time, show your work, and double-check your answers. Equation solving might seem tricky at first, but with practice, you'll become a pro in no time. The steps we used today can be applied to many other equations, so keep practicing and building your skills. You've got this! So, the next time you encounter an equation that looks a little scary, remember the steps we've covered today. Distribute, combine like terms, isolate the variable term, solve for the variable, and always, always check your solution. You'll be amazed at how quickly you can master equation solving. Keep up the great work, and we'll see you next time for more math adventures! Now go out there and conquer those equations!