Solving Equations: A Step-by-Step Guide

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Hey there, math enthusiasts! Ready to dive into the world of equations and uncover the secrets of finding their roots? Don't worry, it's not as scary as it sounds! In fact, with a little patience and a clear understanding of the steps, you'll be solving these problems like a pro. This guide will walk you through solving the equations, breaking down each step to ensure you grasp the concepts. Let's get started! We're going to break down how to find the roots of the equations you've provided. The process involves isolating the variable (in this case, 'x') on one side of the equation. This is achieved by performing the same operations on both sides to maintain the balance.

Equation 1: (x - 67.2) + 4.2 = 17.2 - 20.1

Let's get down to business and solve the first equation. We will be looking at how to solve the equation (x - 67.2) + 4.2 = 17.2 - 20.1. This is a great example to start with, since the process is quite manageable. Our aim is to isolate 'x' on one side of the equation. This involves a couple of simple steps: simplifying both sides of the equation where possible, and then moving the constant terms to one side.

First things first, simplify the right side of the equation. Do the arithmetic. We have 17.2 - 20.1 = -2.9. Now our equation looks like this: (x - 67.2) + 4.2 = -2.9. Next, let's simplify the left side. Combining the constants, we get -67.2 + 4.2 = -63. So, the equation becomes x - 63 = -2.9.

Now, to isolate 'x', we need to move the -63 to the right side of the equation. To do this, we add 63 to both sides. Remember, whatever you do to one side, you must do to the other to keep things balanced! Adding 63 to both sides, we get: x - 63 + 63 = -2.9 + 63. This simplifies to x = 60.1. And that's it, guys! We've found the root of the first equation. The value of x that satisfies the equation is 60.1. Always double-check your work by substituting the value of 'x' back into the original equation to ensure it holds true. This is a crucial step to avoid any mistakes.

Step-by-Step Solution Breakdown

  1. Original Equation: (x - 67.2) + 4.2 = 17.2 - 20.1
  2. Simplify Right Side: (x - 67.2) + 4.2 = -2.9
  3. Simplify Left Side: x - 63 = -2.9
  4. Isolate x: x = -2.9 + 63
  5. Solution: x = 60.1

Equation 2: -100.9 - (47.24 - x) = 5.82 - 90

Alright, let's tackle the second equation: -100.9 - (47.24 - x) = 5.82 - 90. This one involves a little more caution due to the parentheses and the negative signs, but don't sweat it! We'll go through it step by step. Our goal, as always, is to isolate 'x'. First, we'll simplify both sides of the equation. Then, we will deal with the parentheses, which are like the little hurdles in the equation. After that, we gather all the 'x' terms and constant terms. Let's make it happen!

First, let's simplify the right side of the equation: 5.82 - 90 = -84.18. Now the equation looks like this: -100.9 - (47.24 - x) = -84.18. Next, we need to address the parentheses. Remember, the negative sign in front of the parentheses means we have to distribute it to each term inside. So, -(47.24 - x) becomes -47.24 + x. Our equation is now: -100.9 - 47.24 + x = -84.18.

Next, combine the constants on the left side: -100.9 - 47.24 = -148.14. So, we get: -148.14 + x = -84.18. Now we just need to isolate 'x'. To do that, add 148.14 to both sides of the equation: -148.14 + x + 148.14 = -84.18 + 148.14. This simplifies to x = 63.96. We've cracked it! The solution to the second equation is x = 63.96. Again, feel free to plug this value back into the original equation to verify that it works. Practice makes perfect, and with each equation you solve, you'll become more confident in your abilities. Keep up the great work!

Step-by-Step Solution Breakdown

  1. Original Equation: -100.9 - (47.24 - x) = 5.82 - 90
  2. Simplify Right Side: -100.9 - (47.24 - x) = -84.18
  3. Distribute the Negative: -100.9 - 47.24 + x = -84.18
  4. Combine Constants on Left Side: -148.14 + x = -84.18
  5. Isolate x: x = -84.18 + 148.14
  6. Solution: x = 63.96

Tips and Tricks for Solving Equations

Here are some helpful tips to keep in mind as you work through equations. These will help you to be more consistent and accurate. You might also find these to be useful for complex problems, or ones that you make on your own. Remember to always double-check your work, and don’t be afraid to take breaks when you need them. Solving equations can be fun when you understand it!

  • Simplify First: Always try to simplify both sides of the equation as much as possible before starting to isolate the variable. Combine like terms and perform any arithmetic operations. This makes the subsequent steps easier.
  • Balance is Key: Remember that any operation you perform on one side of the equation must also be done on the other side. This ensures that the equation remains balanced.
  • Parentheses: Pay close attention to parentheses and negative signs. Distribute negative signs carefully. Don't rush through this step, it is where many errors occur.
  • Double-Check Your Work: After you find the solution, plug the value back into the original equation to verify that it is correct. This is the best way to catch any calculation errors.
  • Practice Regularly: The more you practice, the more comfortable and confident you will become with solving equations. Work through various examples, starting with simpler problems and gradually increasing the complexity. Seek help when needed!

Conclusion

Congratulations! You've successfully navigated the process of solving these equations. Remember, finding the roots of equations is a fundamental skill in mathematics, and with practice, you'll become more proficient. Keep up the great work, and don't be afraid to challenge yourself with more complex problems. You got this!