Step-by-Step Guide Solving 4(x+2)-5=2(x-1)+7

Hey everyone! Today, we're going to break down how to solve the equation 4(x+2)-5=2(x-1)+7. If you've ever felt a bit lost when tackling these kinds of problems, don't worry – we'll go through each step nice and slowly so you can nail it every time. Solving equations is a fundamental skill in algebra, and mastering it opens the door to more advanced math concepts. This guide is designed to be super clear and helpful, whether you're a student brushing up on your skills or just someone who loves a good math challenge. So, grab your pencil and paper, and let's get started!

Understanding the Basics of Algebraic Equations

Before we dive into the specifics of solving our equation, let's cover some essential basics. Algebraic equations are mathematical statements that assert the equality of two expressions. They contain variables, which are symbols (usually letters like x, y, or z) that represent unknown quantities. The main goal when solving an equation is to find the value of the variable that makes the equation true. Think of it like a puzzle where you need to figure out what number fits perfectly. Key components of an equation include terms, coefficients, constants, and operations. Terms are the individual parts of an expression separated by plus or minus signs. For example, in the expression 4x + 8, 4x and 8 are terms. Coefficients are the numbers that multiply the variables (like the 4 in 4x). Constants are standalone numbers without any variables attached (like the 8 in 4x + 8). Operations, of course, are things like addition, subtraction, multiplication, and division. Understanding these building blocks is crucial because they form the foundation of all algebraic manipulations. When you approach an equation, recognizing these components helps you decide on the best strategy to isolate the variable and find its value. We often use inverse operations to undo operations and simplify the equation. For example, if an equation involves adding a number to the variable, we subtract that number from both sides to balance the equation and isolate the variable. The golden rule of solving equations is that whatever operation you perform on one side, you must perform on the other side to maintain equality. This keeps the equation balanced and ensures that you're moving closer to the correct solution. As we move forward, remember these basics, and you'll find that even complex equations become much easier to handle. Trust me, guys, once you get these basics down, you’ll be solving equations like a pro!

Step 1: Distribute Where Necessary

The first step in solving the equation 4(x+2)-5=2(x-1)+7 is to handle any distribution. Distribution, or the distributive property, is a fundamental concept in algebra that allows us to simplify expressions involving parentheses. The distributive property states that a(b+c) = ab + ac. In simpler terms, it means you multiply the term outside the parentheses by each term inside the parentheses. Looking at our equation, 4(x+2)-5=2(x-1)+7, we have two instances where we need to distribute: the 4 multiplying (x+2) and the 2 multiplying (x-1). Let’s tackle the first one: 4(x+2). Using the distributive property, we multiply 4 by x, which gives us 4x, and then we multiply 4 by 2, which gives us 8. So, 4(x+2) simplifies to 4x + 8. Now, let's move on to the second distribution: 2(x-1). Similarly, we multiply 2 by x, which gives us 2x, and then we multiply 2 by -1, which gives us -2. So, 2(x-1) simplifies to 2x - 2. By applying the distributive property, we've eliminated the parentheses, making the equation easier to work with. Now our equation looks like this: 4x + 8 - 5 = 2x - 2 + 7. This is a significant improvement because we've transformed the equation into a form where we can combine like terms and further simplify it. Guys, remember that mastering distribution is crucial. It's a skill you'll use constantly in algebra and beyond. It’s like having a superpower that lets you break down complex expressions into simpler, more manageable parts. Make sure you feel comfortable with this step before moving on, because it sets the stage for everything else we'll do in solving the equation.

Step 2: Combine Like Terms

After distributing, the next crucial step is to combine like terms. This process simplifies the equation by grouping together terms that have the same variable and exponent (like terms) and constants (numbers without variables). Looking at our equation after distribution, which is 4x + 8 - 5 = 2x - 2 + 7, we can identify the like terms on each side. On the left side, we have the constant terms 8 and -5. Combining these gives us 8 - 5 = 3. So, the left side of the equation simplifies to 4x + 3. On the right side, we have the constant terms -2 and +7. Combining these gives us -2 + 7 = 5. So, the right side of the equation simplifies to 2x + 5. Now, our equation looks much cleaner: 4x + 3 = 2x + 5. This simplified form makes it easier to see the next steps we need to take to isolate the variable. Combining like terms is like tidying up a messy room – it makes everything clearer and more manageable. By grouping similar terms together, we reduce the complexity of the equation, making it easier to see the path to the solution. Remember, you can only combine terms that are “like” each other. That means terms with the same variable raised to the same power (e.g., 4x and 2x) and constants (e.g., 3 and 5) can be combined, but you can't combine terms like 4x and 3 because one has a variable and the other doesn't. Mastering this step is super important, guys. It's a fundamental skill that will help you solve a wide range of algebraic equations more efficiently. When you get good at combining like terms, you’ll notice that even complicated-looking equations start to feel a lot less intimidating.

Step 3: Isolate the Variable Term

Now that we've simplified the equation to 4x + 3 = 2x + 5, the next step is to isolate the variable term. This means getting all the terms with x on one side of the equation and all the constant terms on the other side. To do this, we need to use inverse operations to move terms across the equals sign. A common strategy is to move the variable term with the smaller coefficient first. In our equation, we have 4x on the left side and 2x on the right side. Since 2 is smaller than 4, we'll subtract 2x from both sides of the equation. This gives us: 4x - 2x + 3 = 2x - 2x + 5. Simplifying this, we get 2x + 3 = 5. Now, we have the variable term (2x) isolated on the left side, but we still have the constant term (+3) on the same side. To move the constant term to the other side, we'll subtract 3 from both sides of the equation. This gives us: 2x + 3 - 3 = 5 - 3. Simplifying this, we get 2x = 2. At this point, we've successfully isolated the variable term on one side and the constant term on the other. The equation is now in a form where we can easily solve for x. Isolating the variable term is like building a fence around what you want to find – it separates the unknown from the known. By strategically using inverse operations, we can rearrange the equation to get the variable term all by itself, making it much easier to determine its value. Guys, remember that whatever you do to one side of the equation, you must do to the other side to maintain balance. This principle is key to correctly isolating the variable term. Once you've mastered this step, you're well on your way to finding the solution.

Step 4: Solve for the Variable

After isolating the variable term, we're in the final stretch! Our equation is now 2x = 2. To solve for the variable, which in this case is x, we need to get x completely by itself on one side of the equation. This usually involves one more inverse operation. In our equation, x is being multiplied by 2. To undo this multiplication, we need to divide both sides of the equation by 2. This gives us: (2x) / 2 = 2 / 2. Simplifying this, we get x = 1. And there you have it! We've solved for x, and the solution is x = 1. This means that the value of x that makes the original equation true is 1. Solving for the variable is like the grand finale of the equation-solving process. It's the moment when you finally uncover the unknown value you've been searching for. By using the appropriate inverse operation, you can undo the last remaining operation and reveal the value of the variable. Remember, guys, the key is to think about what operation is being applied to the variable and then use the opposite operation to isolate it. If the variable is being multiplied, you divide; if it's being added, you subtract, and so on. Once you get the hang of this, solving for the variable becomes almost automatic. It’s like the final piece of a puzzle clicking into place.

Step 5: Check Your Solution

We've solved the equation and found that x = 1, but it's always a good idea to check your solution. This ensures that the value we found actually makes the original equation true. To check our solution, we substitute x = 1 back into the original equation: 4(x+2)-5=2(x-1)+7. Substituting x = 1, we get: 4(1+2)-5=2(1-1)+7. Now, we simplify both sides of the equation separately. On the left side, we have: 4(1+2)-5 = 4(3) - 5 = 12 - 5 = 7. On the right side, we have: 2(1-1)+7 = 2(0) + 7 = 0 + 7 = 7. Since both sides of the equation simplify to 7, our solution x = 1 is correct! Checking your solution is like double-checking your work on a test – it gives you confidence that you've got the right answer. By substituting your solution back into the original equation, you can verify that both sides of the equation are equal. If they're not, it means you've made a mistake somewhere along the way, and you need to go back and review your steps. Guys, don't skip this step! It's a crucial part of the problem-solving process. It not only helps you catch errors but also reinforces your understanding of the equation and the solution. Think of it as the final seal of approval on your hard work. When you check your solution and it works, you know you’ve nailed it!

Conclusion

Alright, we've walked through solving the equation 4(x+2)-5=2(x-1)+7 step by step, and we found that x = 1. We covered distributing, combining like terms, isolating the variable term, solving for the variable, and, importantly, checking our solution. By mastering these steps, you'll be well-equipped to tackle all sorts of algebraic equations. Remember, guys, practice makes perfect! The more equations you solve, the more comfortable you'll become with the process. So, keep practicing, stay patient, and don't be afraid to ask for help when you need it. Solving equations is a fundamental skill in math, and it's one that will serve you well in many areas of life. Keep up the great work, and you'll be solving equations like a pro in no time!