Solving Equations: Step-by-Step Guide

Hey math enthusiasts! Let's dive into solving the equation $2(2x - 5) = 3x + x - 2x$. This might look a bit intimidating at first, but trust me, with a few simple steps, we can crack this problem like pros. We'll break it down into easy-to-follow chunks, making sure you understand every move. Ready to get started?

Step-by-Step Solution Breakdown

Alright, guys, here's how we're going to solve this equation step-by-step. It's all about staying organized and keeping track of each operation. Don't worry, I'll walk you through every single detail, so you won't miss a thing. The main goal here is to isolate x on one side of the equation. We do this by simplifying both sides and then using inverse operations to get x all by itself. Sound good? Let's get to work!

First things first, we need to simplify both sides of the equation. This will make it easier to manage and less cluttered. We'll start with the left side. Notice the parentheses? We'll need to distribute the 2 across both terms inside those parentheses. That means multiplying the 2 by both 2x and -5. On the right side, we'll combine the like terms involving x. This involves adding and subtracting the coefficients (the numbers in front of x). By combining these terms, we can make the equation much simpler.

Expanding the Left Side

Let's start by expanding the left side of the equation. We have $2(2x - 5)$. To expand this, we're going to use the distributive property. This means we multiply the 2 by each term inside the parentheses. So, we multiply 2 by 2x, which gives us 4x, and we multiply 2 by -5, which gives us -10. Therefore, the left side of the equation becomes 4x - 10. Remember, the distributive property is your friend here! Always make sure you multiply the outside number by every term inside the parentheses. This is a very common place to make a mistake, so always double-check your work!

Simplifying the Right Side

Now, let's simplify the right side of the equation, which is $3x + x - 2x$. Here, we're dealing with like terms, which means terms that have the same variable (x in this case). To simplify, we'll combine these terms. First, we have 3x. Then we have + x, which is the same as +1x. So, $3x + x$ becomes 4x. Finally, we have -2x. So we subtract 2x from 4x, giving us 2x. Therefore, the right side of the equation simplifies to 2x. Always be careful with the signs – positive and negative signs are crucial and can change the answer. Keep them in mind during all steps, and you’ll do great!

The Transformed Equation

After simplifying both sides, our original equation, $2(2x - 5) = 3x + x - 2x$, transforms into a new, more manageable form. Specifically, we now have 4x - 10 = 2x. See how much cleaner this looks? This transformation is a critical step because it sets us up for isolating x. By simplifying, we reduce the complexity and make the equation easier to solve. The aim of this reduction is to get all the x terms on one side of the equation and all the constant terms (numbers without an x) on the other side. This is the foundation upon which we can find the value of x.

Now that we've got this simplified version, our next goal is to isolate the x terms on one side and the constants on the other. This usually means moving terms from one side to the other. To keep things balanced, whatever you do to one side of the equation, you must also do to the other side. This ensures that the equation remains valid throughout the solving process. Think of it like a seesaw: both sides must be balanced to ensure the equation stays true. Let’s do it!

Isolating the Variable

Our next step is to isolate the variable x. To do this, we want to get all the x terms on one side of the equation and the constant terms on the other. We'll start by subtracting 2x from both sides of the equation, 4x - 10 = 2x. This will eliminate the x term on the right side. It gives us 2x - 10 = 0. Then, to isolate x, we'll add 10 to both sides of the equation. This will move the constant term to the right side and leave us with the x term isolated on the left side. What you do to one side of the equation, you must always do to the other side. Think of it like a scale; you need to maintain balance to get an accurate result.

Subtracting 2x from both sides

To start isolating x, we subtract 2x from both sides. We begin with the equation 4x - 10 = 2x. Subtracting 2x from the left side, we get 4x - 2x - 10, which simplifies to 2x - 10. Subtracting 2x from the right side leaves us with 0. Thus, our equation becomes 2x - 10 = 0. This step brings us closer to isolating the variable x by consolidating all x terms on one side. This is all about getting the x terms together so we can solve for x easier.

Adding 10 to both sides

Now, let's add 10 to both sides of the equation 2x - 10 = 0. On the left side, we have 2x - 10 + 10, which simplifies to 2x. On the right side, we have 0 + 10, which equals 10. So, our equation becomes 2x = 10. This isolates the term with x on the left side and moves the constant to the right side, bringing us one step closer to solving for x. Remember that every step should have a purpose to reach our ultimate goal.

Solving for x

We're in the home stretch, folks! Now that we've isolated the x term, the final step is to solve for x. Our current equation is 2x = 10. To find the value of x, we need to get x all by itself. We do this by dividing both sides of the equation by the coefficient of x, which is 2. This cancels out the 2 on the left side, leaving us with x. On the right side, we divide 10 by 2, which gives us 5. Therefore, the solution to the equation is x = 5. That’s it! We made it! The final step is always straightforward. The most important thing is to make sure you have the basics down and understand what you are doing in each step.

Dividing both sides by 2

To solve for x, we divide both sides of the equation 2x = 10 by 2. On the left side, 2x divided by 2 gives us x. On the right side, 10 divided by 2 gives us 5. Therefore, we get x = 5. This is our final answer! We've successfully isolated x and found its value. Always remember to perform the same operation on both sides of the equation to keep it balanced. It’s a very important rule for all calculations.

The Final Answer

So, after all the steps, the solution to the equation $2(2x - 5) = 3x + x - 2x$ is x = 5. Congratulations! You've successfully solved the equation. Always double-check your work, particularly when dealing with negative signs and simplifying expressions, to ensure accuracy. If you follow each step and pay attention to detail, you will solve any equation. This is one of the most important concepts in mathematics, and now you have the tools to master it.

That's it, guys! We hope this detailed guide made solving this equation super easy for you. Keep practicing, and you'll become a pro at solving equations in no time! Keep practicing, and you will achieve your goals! If you have any more questions, feel free to ask!"