Solving For X: A Step-by-Step Guide

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Solving for X: A Step-by-Step Guide

Hey everyone! Today, we're diving into a classic algebra problem: solving for x. Specifically, we're going to break down how to solve the equation: 4x−47=x−72\frac{4 x-4}{7}=\frac{x-7}{2}. This might look a little intimidating at first, but trust me, it's totally manageable. We'll walk through each step, making sure you understand the 'why' behind the 'how.' Let's get started!

Understanding the Basics: Equations and Variables

Before we jump into the equation, let's quickly recap some fundamental concepts. An equation is a mathematical statement that asserts the equality of two expressions. It's essentially a balance, like a seesaw. Both sides of the equation must always be equal. In our equation, the left side (4x−47\frac{4 x-4}{7}) is equal to the right side (x−72\frac{x-7}{2}). Our goal is to find the value of x that makes this statement true. x is our variable, an unknown value we're trying to figure out. Think of it as a mystery number we need to uncover. The process of finding the value of x is called solving the equation. This involves isolating x on one side of the equation and simplifying the other side to find its numerical value. This is our main goal, to simplify the expression and to solve the equation. We do this by applying various mathematical operations to both sides of the equation while maintaining the balance. We're essentially performing legal moves that maintain the equality until x stands alone. This foundational understanding is key to tackling any algebraic problem.

Let's break down the given equation 4x−47=x−72\frac{4 x-4}{7}=\frac{x-7}{2}. The equation is a fraction equal to a fraction, which can be solved easily by cross-multiplication. The left side is a fraction with 4x - 4 as the numerator and 7 as the denominator. The right side is a fraction with x - 7 as the numerator and 2 as the denominator. The fractions are separated by an equals sign, meaning both sides of the equation are equal to each other. Solving the equation essentially means to find the value of x. The goal is to isolate x on one side of the equation and find its value on the other side. This value of x when substituted into the equation will make both sides equal. The equation can be solved by following a few steps: cross-multiplication, expanding, simplifying, isolating the variable, and finding the final answer.

We will cross-multiply to eliminate the fractions, which means multiplying the numerator of the left side by the denominator of the right side and vice versa. Then we expand the brackets, collect the terms, and isolate the variable x on one side. This process will help you break down complex equations into simpler parts, making them much easier to solve. Always remember to perform the same operations on both sides to maintain the equation's balance. This will help you get the correct answer. Now, let's move on to the actual solution!

Step-by-Step Solution: Unveiling the Value of X

Alright, let's get down to business and solve for x. We'll go through each step carefully, so you can follow along easily. Remember, the key is to perform each operation on both sides of the equation to keep everything balanced.

  1. Cross-Multiplication: This is our first move to get rid of those pesky fractions. We multiply both sides by the denominators of the fractions. Multiply the numerator of the left side (4x - 4) by the denominator of the right side (2). And, multiply the numerator of the right side (x - 7) by the denominator of the left side (7). This gives us:

    2(4x - 4) = 7(x - 7)

    Why do we do this? Because multiplying both sides by the denominators effectively cancels out the fractions, leaving us with a simpler equation to work with.

  2. Expanding the Brackets: Next, we need to expand the brackets. This means multiplying the terms inside the brackets by the number outside. So:

    8x - 8 = 7x - 49

    Why expand? To get rid of the parentheses and combine like terms. This prepares us to isolate x.

  3. Grouping the x Terms: Now, we want to get all the x terms on one side of the equation. Let's subtract 7x from both sides:

    8x - 7x - 8 = 7x - 7x - 49

    This simplifies to:

    x - 8 = -49

    What's the point? By grouping the x terms, we're one step closer to isolating x.

  4. Isolating x: To isolate x, we need to get rid of the -8. We do this by adding 8 to both sides of the equation:

    x - 8 + 8 = -49 + 8

    This simplifies to:

    x = -41

    Almost there! Adding 8 to both sides gets x all by itself.

  5. The Solution: We've done it! We've found that x = -41. This is the solution to our equation. This means that if we substitute -41 for x in the original equation, both sides will be equal. That's how we solve for x.

Verification: Checking Your Answer

It's always a good idea to check your answer. Let's plug -41 back into the original equation to make sure we got it right. Our original equation was 4x−47=x−72\frac{4 x-4}{7}=\frac{x-7}{2}. Let's substitute x = -41:

  1. Substitute: Replace x with -41:

    4(−41)−47=−41−72\frac{4(-41)-4}{7}=\frac{-41-7}{2}

  2. Simplify: Now, simplify each side:

    −164−47=−482\frac{-164-4}{7}=\frac{-48}{2}

    −1687=−24\frac{-168}{7}=-24

    -24 = -24

    The left side equals the right side, so our answer is correct! Always verify your answer by substituting the calculated value back into the original equation. If the left side of the equation equals the right side, you've solved the equation correctly. This is important because it confirms the accuracy of your solution and helps build confidence in your problem-solving skills. By doing this, you are ensuring the validity of your solution. It's an excellent way to catch any errors and solidify your understanding.

Tips and Tricks for Solving Equations

Here are some helpful tips and tricks to make solving equations easier and more efficient:

  • Always Double-Check: Errors can happen, so always double-check your work. Go through each step again to make sure you didn't miss anything. Verify your answer by plugging it back into the original equation. It's a lifesaver. This will help you identify any mistakes made during the process. Practice and consistency are key to mastering equation solving.
  • Simplify First: Before you start moving things around, look for ways to simplify the equation. Combine like terms where possible. The simpler the equation, the easier it is to solve.
  • Write Everything Down: Don't try to do too much in your head. Write down each step clearly. This helps you keep track of your work and makes it easier to spot any errors.
  • Practice Regularly: The more you practice, the better you'll get at solving equations. Work through various examples. This will help you get familiar with different types of equations and the strategies to solve them.
  • Understand the Rules: Make sure you know the basic rules of algebra. These rules are the foundation for solving equations. Ensure that you have a firm grasp of the concepts and are confident in their application.
  • Stay Organized: Keep your work neat and organized. This will prevent confusion and help you avoid making mistakes.

Final Thoughts: Conquering Equations

So there you have it, guys! We've successfully solved for x and simplified the equation 4x−47=x−72\frac{4 x-4}{7}=\frac{x-7}{2}. Remember, the key is to break the problem into smaller steps, stay organized, and double-check your work. Solving equations can seem daunting at first, but with practice and a good understanding of the basics, you'll be solving them like a pro in no time! Always remember that math is a journey, and every problem you solve makes you stronger. Keep practicing and keep learning, and you'll do great. Good luck, and keep those equations in check! Keep practicing and don't be afraid to ask for help when you need it. You got this!