Math Problem: Solve The Equations Step-by-Step
Hey math whizzes! Let's tackle these equations together. We'll break down each problem, step by step, to make sure we understand everything perfectly. Don't worry, it's gonna be fun! We'll start with the first one and move through each part methodically. Get ready to flex those math muscles!
Equation 1: Decoding the First Equation
Alright, guys, let's dive into the first equation: (18.14 - 20.49) - (-3.35). This is where things get interesting. We'll start with the parentheses first, as always, following the order of operations (PEMDAS/BODMAS – remember those?). We have a subtraction inside the first set of parentheses. So, we'll perform that operation. 18.14 - 20.49 equals -2.35. Now, our equation looks like this: -2.35 - (-3.35). Here's where we need to remember a crucial rule: subtracting a negative number is the same as adding its positive counterpart. Therefore, -2.35 - (-3.35) becomes -2.35 + 3.35. When we add these two numbers together, we get 1. Voila! The first part of our first equation simplifies to 1. Not too shabby, right? Next up, we will tackle the second part of the first equation, which involves fractions and a few other operations. We will be using the order of operations as well, as this is the key to solving mathematical problems.
Equation 1 (Continued): Fractions and More
Now, let's tackle the second part of the equation: 3/2 - 5 + 5/3. This involves fractions, and we need to be a little careful. To effectively work with these fractions, it's often easiest to convert them into a common denominator. The least common denominator (LCD) for 2 and 3 is 6. So, let's convert our fractions: 3/2 becomes 9/6 and 5/3 becomes 10/6. Now our equation looks like this: 9/6 - 5 + 10/6. We can simplify this further. First, combine the fractions: 9/6 + 10/6 equals 19/6. Now, the equation is 19/6 - 5. To subtract 5, which we can consider as 5/1, we'll need to rewrite 5 with a denominator of 6. So, 5 becomes 30/6. Now we have 19/6 - 30/6, which equals -11/6. That's a good result. Now we've simplified both parts of the first equation. The first part equals 1. The second part equals -11/6. Depending on the question, we may need to combine these two results, or the original question may be incorrectly written. However, we have solved the individual steps for the problem.
Equation 2: Another Math Challenge
Let's move on to the next equation: (7/5 + 6 - 29/4). This one looks like another good challenge. We will use the same principles and the order of operations. First, let's handle the fractions. We'll need to find a common denominator for 5 and 4, which is 20. So, we convert our fractions: 7/5 becomes 28/20 and 29/4 becomes 145/20. Now our equation is (28/20 + 6 - 145/20). Let's combine those fractions: 28/20 - 145/20 equals -117/20. Now our equation is -117/20 + 6. We can express 6 as 120/20. So, the equation becomes -117/20 + 120/20. Adding these together, we get 3/20. So, the result for the second equation is 3/20. It's a bit more complex, but we've got it!
Equation 2 (Continued): A Quick Check
It is always a good idea to perform a quick check, to ensure the correctness of our solution. We can convert fractions to decimals to see if the equation is more readable. The result of 3/20 is 0.15. The fraction is positive, which implies that our equation and calculations are correct. If we had arrived at a negative fraction, then we would need to check our work. However, in this case, the result is in line with our expectations. Always remember, when solving an equation, the key is to be methodical and check your work at each step. This also applies when solving multiple equations. You will have to do each step of the way, and ensure that each step you are doing is correct. If one of the steps is not correct, you will get the wrong result. Take your time, and enjoy the process. These equations are not supposed to be solved in haste, but rather with dedication, attention to detail and patience.
Equation 3: The Final Countdown
Alright, let's wrap things up with the final equation: 5.5 + 3/2 and (22.19 - 40.2) - (-20.10). This equation has a combination of decimals and fractions. Let's make it our final target. Let's start with 5.5 + 3/2. First, we convert the fraction 3/2 into a decimal, which is 1.5. So, the equation becomes 5.5 + 1.5, which equals 7. Easy, right? Now, let's address the second part: (22.19 - 40.2) - (-20.10). Inside the parentheses, 22.19 - 40.2 equals -18.01. So, now we have -18.01 - (-20.10). Remember the rule? Subtracting a negative is the same as adding. So, we get -18.01 + 20.10, which equals 2.09. That's a good result. Now we've got the simplified answers for both parts. The first part equals 7, and the second part equals 2.09. Depending on the question, we may need to combine these two results, or the original question may be incorrectly written. We have solved the individual parts.
Equation 3 (Continued): Checking the Results
Let's do a quick check to make sure our final equation makes sense. We can convert the results to fractions. 7 can be written as 7/1 and 2.09 can be written as 209/100. This will allow us to see if we have performed the calculations correctly. We can also cross-check our results with a calculator. The process is a good habit. You may have noticed that when solving these equations, we have followed a specific method. Always start with the parentheses, then deal with exponents (if there are any), then multiplication and division (from left to right), and finally addition and subtraction (from left to right). This order is critical for the correct answer. Even a small misstep can lead to a wrong answer, so it's always crucial to stay focused and not rush the process. When dealing with fractions, finding a common denominator is your best friend. It makes the adding and subtracting much easier. Converting fractions to decimals can also help in visualizing the equations. This makes sure that the results you have gotten are not off. Overall, there are multiple techniques to arrive at the same solution. This is what makes solving equations so interesting. Each individual can experiment with different methods, until the one that suits them best is found.
Conclusion: You've Got This!
Fantastic job, everyone! We've successfully navigated through all the equations. We've conquered fractions, decimals, and negative numbers. Remember, practice makes perfect. Keep solving problems, and you'll become math wizards in no time. If there is something that feels unclear, revisit the examples, repeat the steps, and remember the rules. If you continue to practice, you will become a master of solving mathematical equations. Keep up the amazing work!