Solving For X: A Step-by-Step Guide To The Equation
Hey math enthusiasts! Ever stumbled upon an equation that looks like a tangled mess? Don't worry, we've all been there. Today, we're diving deep into a specific equation and breaking down the solution step-by-step. So, grab your pencils, and let's get started!
The Challenge: 4x - 3 + 5 = 2x + 7 - 8x
The equation we're tackling today is: 4x - 3 + 5 = 2x + 7 - 8x. At first glance, it might seem intimidating, but fear not! With a systematic approach, we can conquer this mathematical beast.
Step 1: Simplify Both Sides
The first thing we want to do is simplify each side of the equation. This involves combining like terms. On the left side, we have -3 and +5, which combine to +2. On the right side, we have 2x and -8x, which combine to -6x. This gives us a simplified equation:
4x + 2 = -6x + 7
Simplifying both sides of the equation is a crucial first step. By combining like terms, we reduce the complexity of the equation, making it easier to work with. Think of it as decluttering your workspace before starting a project – a clean equation is a happy equation! We are essentially making the equation more manageable by grouping similar terms together. This process not only simplifies the equation visually but also sets the stage for the next steps in solving for the unknown variable, 'x'. Remember, the goal is to isolate 'x' on one side of the equation, and simplifying is the first step towards achieving that goal. By combining constants and variable terms separately, we create a clearer picture of the relationship between the different components of the equation. This streamlined version of the equation allows us to focus on the core problem – finding the value of 'x' that satisfies the equality.
Step 2: Move the x Terms to One Side
Our goal is to isolate x on one side of the equation. To do this, we need to get all the x terms on the same side. A common strategy is to move the x terms to the side with the larger coefficient. In this case, we have 4x on the left and -6x on the right. To eliminate the -6x on the right, we add 6x to both sides of the equation:
4x + 2 + 6x = -6x + 7 + 6x
This simplifies to:
10x + 2 = 7
Moving the x terms to one side is a pivotal step in solving for x. It’s like gathering all the ingredients you need for a recipe in one place. By bringing all the x terms together, we can then isolate x more easily. We achieve this by performing the same operation on both sides of the equation, ensuring that the equation remains balanced. This principle of equality is fundamental to solving algebraic equations. Adding 6x to both sides cancels out the -6x on the right side, effectively transferring the x term to the left side of the equation. This strategic move consolidates the x terms, paving the way for the next step – isolating x completely. By focusing on this goal of isolating the variable, we are methodically working towards the solution. This step demonstrates the power of algebraic manipulation in simplifying complex equations and bringing us closer to the answer.
Step 3: Isolate the x Term
Now, we want to isolate the x term completely. We have 10x + 2 = 7. To get the 10x term by itself, we need to get rid of the +2. We do this by subtracting 2 from both sides of the equation:
10x + 2 - 2 = 7 - 2
This simplifies to:
10x = 5
Isolating the x term is a crucial step towards unlocking the value of x. It's akin to clearing away the obstacles that stand between you and your destination. In our equation, the '+ 2' is currently hindering the x term from being completely alone on one side. To remove this obstacle, we employ the principle of inverse operations. Since 2 is being added to 10x, we perform the opposite operation – subtraction. By subtracting 2 from both sides of the equation, we maintain the balance and effectively eliminate the '+ 2' from the left side. This leaves us with 10x isolated, which means we are one step closer to finding the value of x. This process of isolating the variable is a fundamental technique in algebra, allowing us to systematically unravel the equation and reveal the solution. It showcases the power of algebraic manipulation in simplifying complex expressions and moving us closer to the final answer. This step is a testament to the methodical approach required to solve equations, where each operation brings us closer to the ultimate goal.
Step 4: Solve for x
Finally, we solve for x by dividing both sides of the equation by the coefficient of x, which is 10:
10x / 10 = 5 / 10
This simplifies to:
x = 1/2
Therefore, the solution to the equation is x = 1/2. We can also express this as a decimal: x = 0.5
Solving for x is the final, triumphant step in our mathematical journey! It's like reaching the summit after a challenging climb, or discovering the hidden treasure at the end of a quest. We've meticulously simplified the equation, isolated the x term, and now we're ready to reveal its value. The x term is currently multiplied by 10, so to undo this multiplication and isolate x, we perform the inverse operation – division. By dividing both sides of the equation by 10, we maintain the balance and effectively separate x from its coefficient. This division unveils the value of x, which in this case is 1/2 or 0.5. This moment of revelation is the culmination of all our efforts, the tangible result of our methodical approach. Solving for x not only provides the answer to the equation but also reinforces the power of algebraic manipulation in unraveling complex mathematical problems. It's a testament to the elegance and precision of mathematics, where each step logically leads to the final solution. This achievement underscores the importance of persistence and a systematic approach in tackling mathematical challenges.
The Answer
So, the correct answer is D. x = 1/2
Let's Recap
To recap, we solved the equation 4x - 3 + 5 = 2x + 7 - 8x by following these steps:
- Simplified both sides.
- Moved the x terms to one side.
- Isolated the x term.
- Solved for x.
Final Thoughts
Solving equations can be a fun and rewarding experience. Remember to take it step by step, and don't be afraid to ask for help if you get stuck. Keep practicing, and you'll become a math whiz in no time!