Solving -3(x-1) = X-5: A Step-by-Step Guide

Hey math enthusiasts! Ever stumbled upon an equation that looks a bit intimidating at first glance? Today, we're going to break down a common type of algebraic equation and show you exactly how to solve it. We’ll be tackling the equation -3(x-1) = x-5. Don't worry, it's not as scary as it looks! We'll go through each step methodically, so you can follow along and master these types of problems. Whether you're a student prepping for an exam or just brushing up on your algebra skills, this guide is for you. Let’s dive in and conquer this equation together!

Understanding the Equation

Before we jump into solving, let's make sure we understand what the equation is telling us. The equation -3(x-1) = x-5 is a linear equation, which means it involves a variable (in this case, 'x') raised to the power of 1. Our goal is to find the value of 'x' that makes the equation true. In simpler terms, we want to find the number that, when plugged in for 'x', will make both sides of the equation equal. This involves using algebraic manipulations to isolate 'x' on one side of the equation. We'll use properties of equality, like the distributive property and combining like terms, to simplify the equation step by step. Remember, the key to solving any equation is to maintain balance, so whatever operation we perform on one side, we must also perform on the other side. So, grab your pencils, and let's get started on this journey to find the value of 'x'!

The Distributive Property: Key to Unlocking the Equation

The first step in solving the equation -3(x-1) = x-5 is to apply the distributive property. This property states that for any numbers a, b, and c, a(b + c) = ab + ac. In our equation, we need to distribute the -3 across the terms inside the parentheses (x and -1). This means we'll multiply -3 by both 'x' and '-1'. Understanding and correctly applying the distributive property is crucial, as it simplifies the equation and allows us to proceed with solving for 'x'. Many algebraic errors occur during this step, so pay close attention and double-check your work. We're essentially breaking down the complex part of the equation into smaller, more manageable pieces. Once we've distributed the -3, the equation will look different, and we'll be one step closer to isolating 'x'. So, let's get that distributive property working and transform our equation!

Step-by-Step Application of the Distributive Property

Okay, let’s put the distributive property into action! We have -3(x-1) on the left side of our equation. To distribute the -3, we multiply it by 'x' and then by '-1'. So, -3 multiplied by 'x' is -3x, and -3 multiplied by -1 is +3 (remember, a negative times a negative is a positive). Therefore, after applying the distributive property, -3(x-1) becomes -3x + 3. Now, let's rewrite our entire equation with this simplification: -3x + 3 = x - 5. See how we've transformed the equation? We've eliminated the parentheses and created a more streamlined equation to work with. This step is a crucial foundation for the rest of the solution, so it's important to be confident that you've distributed correctly. We're making progress towards isolating 'x', and the equation is becoming less intimidating with each step. Let's move on to the next phase of our solution!

Isolating the Variable 'x'

Now that we've applied the distributive property, our equation looks like this: -3x + 3 = x - 5. Our next goal is to isolate the variable 'x' on one side of the equation. This means we want to get all the terms with 'x' on one side and all the constant terms (the numbers without 'x') on the other side. To do this, we'll use the properties of equality, which allow us to add or subtract the same value from both sides of the equation without changing its balance. The key here is to strategically move terms across the equals sign by performing the opposite operation. For instance, if we want to get rid of a term being added on one side, we subtract it from both sides. This process of isolating 'x' is like solving a puzzle, where each move brings us closer to the final solution. So, let's start moving terms and get 'x' all by itself!

Combining Like Terms: The Art of Simplification

Before we can completely isolate 'x', we need to combine like terms. Looking at our equation, -3x + 3 = x - 5, we have 'x' terms on both sides (-3x and x) and constant terms on both sides (3 and -5). To combine like terms, we'll perform operations to move the 'x' terms to one side and the constant terms to the other. A common strategy is to move the 'x' terms to the side with the larger coefficient (in this case, the left side since -3 is less than 1) and the constant terms to the opposite side. This helps avoid dealing with negative coefficients, but it's not a strict rule. Remember, whatever operation we perform on one side, we must also perform on the other to maintain the equation's balance. Combining like terms simplifies the equation and makes it easier to solve for 'x'. Let's start combining and watch our equation transform!

Moving 'x' Terms to One Side

Let's start by moving the 'x' terms to one side of the equation. We have -3x + 3 = x - 5. To get rid of the 'x' on the right side, we'll subtract 'x' from both sides of the equation. This gives us: -3x + 3 - x = x - 5 - x. Simplifying this, we get -4x + 3 = -5. Notice how the 'x' term has disappeared from the right side, and we now have -4x on the left side. We're making great progress in isolating 'x'! This step is crucial because it consolidates all the 'x' terms into one expression. By subtracting 'x' from both sides, we've effectively shifted the 'x' term from the right to the left, while maintaining the balance of the equation. Now that we have all the 'x' terms on one side, let's focus on moving the constant terms to the other side. We're one step closer to finding the value of 'x'!

Shifting Constant Terms: The Final Push

Now that we have -4x + 3 = -5, it's time to move the constant terms to the right side of the equation. We want to isolate the term with 'x' (-4x) on the left side. To do this, we need to get rid of the +3 on the left side. We'll subtract 3 from both sides of the equation: -4x + 3 - 3 = -5 - 3. This simplifies to -4x = -8. Look at that! We've successfully isolated the 'x' term on one side and moved all the constant terms to the other. We're in the home stretch now. This step of moving constant terms is the final push before we can directly solve for 'x'. By subtracting 3 from both sides, we've balanced the equation and cleared the way for the final calculation. With -4x = -8, we're just one operation away from discovering the value of 'x'. Let's finish strong and solve for 'x'!

Solving for 'x'

We've arrived at the equation -4x = -8. This is the moment we've been working towards – solving for 'x'! To isolate 'x', we need to undo the multiplication that's happening between -4 and 'x'. The opposite operation of multiplication is division. So, we'll divide both sides of the equation by -4. Remember, whatever we do to one side, we must do to the other to maintain balance. This step is the culmination of all our previous efforts. We've simplified the equation, moved terms around, and now we're ready to find the exact value of 'x'. Dividing by -4 will free 'x' from its coefficient and reveal the solution. Let's perform this final operation and uncover the value of 'x'!

Performing the Division: Unveiling the Solution

Let's perform the division and find the solution! We have -4x = -8. To isolate 'x', we divide both sides by -4: (-4x) / -4 = (-8) / -4. On the left side, -4 divided by -4 cancels out, leaving us with just 'x'. On the right side, -8 divided by -4 equals 2 (remember, a negative divided by a negative is a positive). So, our solution is x = 2. We've done it! We've successfully solved the equation and found the value of 'x'. This step is the grand finale, the moment where all our hard work pays off. By dividing both sides by -4, we've revealed the value of 'x' that makes the original equation true. Now that we have our solution, let's make sure it's correct by checking our work.

Checking the Solution

Now that we've found that x = 2, it's crucial to check our solution. This ensures that our answer is correct and that we haven't made any mistakes along the way. To check, we'll substitute x = 2 back into the original equation, -3(x-1) = x-5, and see if both sides of the equation are equal. If they are, then our solution is correct. Checking our solution is like double-checking our work on a test – it's a vital step in the problem-solving process. It gives us confidence in our answer and helps us catch any errors we might have made. So, let's plug in x = 2 and verify that our solution holds true!

Substituting 'x = 2' into the Original Equation

Let's substitute x = 2 into our original equation, -3(x-1) = x-5. Replacing 'x' with 2, we get: -3(2-1) = 2-5. Now, let's simplify both sides of the equation. On the left side, we first evaluate the expression inside the parentheses: 2 - 1 = 1. So, we have -3(1) = 2 - 5. Multiplying -3 by 1 gives us -3, so the left side becomes -3. On the right side, 2 - 5 = -3. So, our equation now reads: -3 = -3. This is a true statement! Both sides of the equation are equal, which means our solution, x = 2, is correct. This step of substitution is the final confirmation that we've solved the equation accurately. We can now confidently say that x = 2 is the solution to the equation -3(x-1) = x-5.

Conclusion: Mastering Algebraic Equations

Great job, everyone! We've successfully navigated the equation -3(x-1) = x-5 and found the solution, x = 2. We started by understanding the equation, applied the distributive property, isolated the variable 'x', solved for 'x', and then checked our solution. This step-by-step approach is key to mastering algebraic equations. Remember, the most important thing is to understand the underlying principles and apply them methodically. Practice makes perfect, so the more equations you solve, the more confident you'll become. Algebraic equations are a fundamental part of mathematics, and being able to solve them is a valuable skill. So, keep practicing, keep learning, and you'll be solving even more complex equations in no time! You've got this!