Finding The Number: Solving The Equation Step-by-Step

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Finding the Number: Solving the Equation Step-by-Step

Hey guys! Let's dive into a fun math problem. We're going to break down how to solve an equation that looks a little tricky at first glance. The original problem is: "El triple de un número, disminuido en 19, es igual a 53. Determina el valor de dicho número." which translates to "The triple of a number, decreased by 19, is equal to 53. Determine the value of that number." Don't worry, it's easier than it sounds! We'll go step-by-step, so you can totally nail it. We will be discussing the mathematical concepts required to solve it and understand it from the basics. So buckle up, because we're about to become math wizards! We'll start with understanding the words and translating them into a mathematical equation. Understanding the words is the key, this is very important because once you understand it, it is as simple as it sounds.

Breaking Down the Problem

Okay, so the first thing we need to do is understand what the problem is asking. We're looking for a number, right? Let's call this number "x." Now, let's look at the sentence again, "The triple of a number, decreased by 19, is equal to 53." This is where we break it down piece by piece.

  • "The triple of a number": This means we're multiplying our number "x" by 3. In math, this is written as 3x.
  • "decreased by 19": This means we're subtracting 19 from our 3x. So, it becomes 3x - 19.
  • "is equal to 53": This means our whole expression, 3x - 19, equals 53. In math, we write this as 3x - 19 = 53.

Now, we have our equation: 3x - 19 = 53. See? Not so scary once we break it down. We've taken a word problem and translated it into a mathematical equation. This is a very common approach to solving mathematical problems in the real world. This translation is the key to solving the equation, it is important to remember what each of the words or phrases means. The concept is that words in a sentence can be translated into mathematical symbols and that mathematical symbols are a shorter way of representing a concept or problem. Remember that in math, a simple change of a symbol can change the whole meaning of the equation.

The Importance of Understanding the Equation

Understanding the equation is the foundation for solving any math problem. It’s like having the right map before you start a journey. Without it, you might get lost. So, if we break the equation into smaller pieces we can easily find the end. The equation, as we have already seen, is a mathematical statement that asserts the equality of two expressions. When we understand the equation, we can start with the basics of what we already know and then add the new concepts we learned. This builds a mental image of what is to be solved. Let's make sure we're on the right track. The equation 3x - 19 = 53 states that if we take a number, multiply it by 3 (making it the triple), and then subtract 19 from that result, we'll get 53. Our goal is to find the original number (x). Once we find the value of x, we can say that we've solved the equation. Now that we understand the equation we can start working on finding the missing value or x. The goal of solving the equation is always to find the missing variable.

Solving for "x"

Alright, now for the fun part – solving the equation! We have 3x - 19 = 53. Our goal is to get "x" by itself on one side of the equation. Here's how we do it, step-by-step:

  1. Get rid of the -19: To do this, we need to do the opposite operation. The opposite of subtracting 19 is adding 19. So, we add 19 to both sides of the equation. This is super important – whatever you do to one side of the equation, you MUST do to the other side to keep it balanced. So, it becomes: 3x - 19 + 19 = 53 + 19.
  2. Simplify: On the left side, -19 + 19 cancels out, leaving us with 3x. On the right side, 53 + 19 = 72. Our equation now looks like this: 3x = 72.
  3. Isolate "x": We have 3x, which means 3 multiplied by x. To get "x" by itself, we do the opposite of multiplying by 3, which is dividing by 3. We divide both sides of the equation by 3. So, it becomes: 3x / 3 = 72 / 3.
  4. Solve for "x": On the left side, 3 / 3 cancels out, leaving us with just "x." On the right side, 72 / 3 = 24. Therefore, x = 24.

And there you have it! We've solved for "x". That means the value of the number is 24.

Explanation of the Steps

Let's break down each step and why it works. When we are solving an equation like 3x - 19 = 53, the main goal is to isolate the variable (in this case, x) on one side of the equation. This means getting x by itself. To achieve this, we use the principles of inverse operations. An inverse operation is the operation that reverses the effect of another operation. For example, addition and subtraction are inverse operations. Similarly, multiplication and division are inverse operations. In our first step, we added 19 to both sides of the equation because it is the inverse operation of subtracting 19. This eliminated the -19 on the left side. The basic concept is to get x by itself. Then, to get x by itself, we needed to divide both sides by 3. This is because it is the inverse operation of multiplying x by 3. This left x isolated on the left side of the equation, and we were left with x = 24. Each of these steps maintains the balance of the equation, ensuring that the equality holds true. By consistently applying these principles, we can solve for any variable in any linear equation.

Checking Your Work

Always check your answer! It's super easy to do and helps ensure you got it right. Let's go back to our original problem and plug in our answer (x = 24) to see if it works.

Remember: "The triple of a number, decreased by 19, is equal to 53." Let's substitute 24 for the number:

  • Triple of 24: 3 * 24 = 72
  • Decreased by 19: 72 - 19 = 53

And guess what? It works! 53 = 53. Our answer is correct. This confirmation is crucial. When you perform this step, you can make sure that all of the calculations are correct. If the numbers don't match, you know that there has been an error. This is important to ensure that you are on the right track and that you are not missing any steps. If there is an error, this will help you identify exactly where the error is.

The Importance of Verification

Verification is an essential aspect of solving any mathematical problem. It's the step where you check your solution to ensure its accuracy. In the context of our problem, verification involves substituting the value of x (which is 24) back into the original equation and confirming that it satisfies the equation. The process provides confidence in the answer. To verify the answer we need to substitute the result we found with x and evaluate the equation. In our case, the original equation was 3x - 19 = 53. Substituting x = 24, we get 3(24) - 19 = 72 - 19 = 53. This verifies that our solution is correct. In simple words, if the value we found for x does not make the equation true, then we know that our answer is not correct. It means we have to go back and check our steps. This process ensures that you have successfully solved the problem. Without checking, there's always a chance of an error, and verification reduces this risk. This is the cornerstone of problem-solving. It builds the skill and accuracy needed to excel in mathematics.

More Examples

Want to practice more? Let's try another one. "Double a number, and then add 10, the result is 30." Can you solve for the number?

  • Translate: 2x + 10 = 30
  • Subtract 10 from both sides: 2x = 20
  • Divide both sides by 2: x = 10

Check: 2 * 10 + 10 = 30. Correct!

Here's another one: "Half of a number, plus 5, equals 11." Can you solve for the number?

  • Translate: x/2 + 5 = 11
  • Subtract 5 from both sides: x/2 = 6
  • Multiply both sides by 2: x = 12

Check: 12/2 + 5 = 11. Correct!

Practice Makes Perfect

Practicing different types of problems is very important. This is how you will be able to master the concepts and methods. Keep practicing. This will help you become a master. The more examples you solve, the more comfortable you will become with translating words into equations and solving for the unknown variable. Each time you solve a new problem, you build upon the previously learned concepts. You are building confidence and problem-solving skills, and also enhancing the ability to think logically. Different types of problems help with the understanding of the concepts. Keep in mind that when practicing, the goal is not only to find the right answer but also to understand the steps involved in arriving at the answer. Therefore, taking a step-by-step approach to each problem is important. If you start to solve many problems, the steps will become easier and easier. This continuous effort will help you to think critically and apply mathematical principles to various scenarios.

Conclusion

Awesome work, guys! We've successfully solved the equation and learned a valuable lesson about breaking down problems step by step. Remember, math is like a puzzle. If you take it piece by piece, you can solve any problem. Keep practicing, and you'll become a math whiz in no time. If you got stuck, go back and review the steps. The more you solve different types of problems, the easier it will be to master the concepts. Remember to always check your answers. This will boost your confidence and help you to become better at math.