Solving For The Unknown The Double Of A Number Plus 5 Equals 23

Hey there, math enthusiasts! Today, we're diving into a classic algebraic problem. We've got a mystery number, and the puzzle goes like this: if you double the number and then add 5, the result is 23. What's the number? Sounds intriguing, right? Let's break it down step by step and uncover the solution together.

Setting Up the Equation: The First Step to Solving the Puzzle

Before we can find our mystery number, we need to translate the words into a mathematical equation. This is a crucial step in algebra, guys, because it allows us to use the power of symbols and operations to solve the problem. Our main keyword here is equation setup, and understanding this process is key to tackling many mathematical challenges.

So, let's represent our unknown number with a variable. A common choice is "x," but you can use any letter you like – "n" for number, "y" for why not? In this case, let’s stick with "x" for simplicity. The problem states that we need to double this number. Doubling something means multiplying it by 2, so we have 2 * x, which we can write more simply as 2x. Next, we add 5 to this doubled number, giving us 2x + 5. And finally, we know that this whole expression equals 23. Putting it all together, we get the equation:

2x + 5 = 23

This equation is the heart of our problem. It's a concise way of expressing the relationship described in the original word problem. The beauty of algebra is that it allows us to manipulate these symbols to isolate the variable and find its value. Now, with our equation in place, we're ready to move on to the next step: solving for "x".

But why is setting up the equation so important? Well, think of it as building the foundation of a house. If the foundation is shaky, the whole structure is at risk. Similarly, if the equation doesn't accurately represent the problem, the solution will be wrong. A correctly set up equation acts as a roadmap, guiding us through the steps needed to arrive at the answer. It's like having a clear destination programmed into your GPS – it ensures you're heading in the right direction.

Another key aspect of equation setup is understanding the order of operations. Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)? This order dictates how we interpret and manipulate mathematical expressions. In our equation, 2x + 5, we understand that the multiplication (2 * x) happens before the addition (+ 5). This understanding is crucial when we start to solve for "x" because we need to reverse these operations in the correct order.

Moreover, the ability to translate word problems into equations is a valuable skill that extends far beyond the classroom. In the real world, we often encounter situations that can be modeled mathematically. Whether it's calculating a budget, determining the optimal price for a product, or analyzing data, the ability to set up equations allows us to make informed decisions. So, mastering this skill isn't just about acing your math test; it's about equipping yourself with a powerful tool for problem-solving in all areas of life. That's why paying attention to the equation setup is so important, guys!

Solving for 'x': Unraveling the Mystery Number

Alright, now that we've got our equation firmly in place (2x + 5 = 23), it's time for the exciting part: solving for "x". This is where we put our algebraic skills to the test and isolate the variable to reveal its value. The main keyword we're focusing on here is solving equations, and it's a fundamental concept in algebra.

The goal of solving equations is to get "x" all by itself on one side of the equation. To do this, we need to undo the operations that are being performed on "x". Remember, we're essentially working backward through the order of operations. In our equation, "x" is being multiplied by 2 and then 5 is being added. So, to isolate "x", we need to first undo the addition and then the multiplication.

The first step is to subtract 5 from both sides of the equation. Why both sides? Because equations are like balanced scales – whatever you do to one side, you must do to the other to maintain the balance. Subtracting 5 from both sides gives us:

2x + 5 - 5 = 23 - 5

This simplifies to:

2x = 18

Notice how we've successfully undone the addition and brought us closer to isolating "x". Now, "x" is only being multiplied by 2. To undo this multiplication, we need to divide both sides of the equation by 2:

2x / 2 = 18 / 2

This simplifies to:

x = 9

And there you have it! We've solved for "x". Our mystery number is 9. Pat yourselves on the back, guys, we just mastered the art of solving equations!

But the process of solving equations isn't just about following steps; it's about understanding why those steps work. Each operation we perform is designed to simplify the equation while maintaining its balance. Think of it as carefully peeling away layers to reveal the core value of "x". We're not just blindly applying rules; we're strategically manipulating the equation to isolate the variable.

Furthermore, solving equations is a skill that builds upon itself. As you encounter more complex equations, the same fundamental principles apply. You'll learn to deal with parentheses, fractions, and even multiple variables, but the core concept of isolating the variable remains the same. It's like learning the basics of a language – once you have the grammar and vocabulary down, you can express yourself in countless ways.

Moreover, the ability to solve equations is crucial in many fields, from science and engineering to finance and economics. Scientists use equations to model the physical world, engineers use them to design structures and machines, and economists use them to analyze markets. Solving equations isn't just an academic exercise; it's a practical skill that empowers you to understand and interact with the world around you. So, keep practicing, guys, and you'll become equation-solving pros in no time!

Verifying the Solution: Ensuring Accuracy and Confidence

We've found our solution, x = 9, but before we declare victory, there's one crucial step remaining: verifying the solution. Our key focus here is solution verification, and it's the safety net that catches any potential errors. It ensures that our answer not only looks right but actually works within the context of the original problem.

Solution verification is like checking your work on a test. You might feel confident about your answer, but taking a few minutes to double-check can catch mistakes you might have missed. In algebra, verifying the solution involves plugging the value we found for "x" back into the original equation and seeing if it holds true. If the equation remains balanced, we can be confident in our answer.

Our original equation was:

2x + 5 = 23

Now, let's substitute "x" with 9:

2 * 9 + 5 = 23

Following the order of operations, we first multiply 2 by 9:

18 + 5 = 23

Then, we add 18 and 5:

23 = 23

Voila! The equation holds true. This confirms that our solution, x = 9, is indeed correct. We can breathe a sigh of relief and confidently move on. We nailed the solution verification, guys!

But why is solution verification so important? Well, think of it as a safeguard against mistakes. We're all human, and even the most experienced mathematicians can make errors. A simple arithmetic mistake, a sign error, or a misapplication of a rule can lead to an incorrect solution. Solution verification is the final line of defense against these errors. It's a quick and easy way to ensure that your answer is accurate.

Moreover, solution verification builds confidence. When you verify your solution and see that it works, you know you've solved the problem correctly. This confidence can be invaluable, especially when tackling more complex problems. It allows you to approach new challenges with a sense of assurance, knowing that you have the skills and the tools to succeed.

In addition to catching errors, solution verification can also help you develop a deeper understanding of the problem. By plugging your solution back into the original equation, you're essentially revisiting the relationships between the variables and the constants. This can provide valuable insights into the problem and help you solidify your understanding of the underlying concepts. It is like reviewing the map after arriving at the destination, ensuring we understood the journey well.

So, always remember to verify your solutions, guys. It's a simple step that can save you time, prevent frustration, and build confidence. It's the mark of a meticulous and successful problem-solver. Solution verification is the cherry on top of a perfectly solved mathematical problem!

Conclusion: The Power of Algebraic Thinking

We've successfully navigated our math problem, guys! We took a word problem, translated it into an equation, solved for the unknown, and verified our solution. This journey highlights the power of algebraic thinking, which is the ability to represent relationships with symbols and manipulate those symbols to find solutions. Our key takeaway here is algebraic thinking, and it's a skill that's valuable far beyond the realm of mathematics.

Throughout this problem, we've used several important algebraic concepts. We learned how to set up an equation, how to isolate a variable, and how to verify our solution. Each of these steps is a building block of algebraic thinking. Setting up the equation allows us to translate a real-world situation into a mathematical model. Solving for the variable allows us to uncover hidden information. And verifying the solution ensures that our answer is accurate and reliable.

But algebraic thinking is more than just a set of skills; it's a way of approaching problems. It involves breaking down complex situations into smaller, more manageable parts. It involves identifying patterns and relationships. And it involves using logic and reasoning to arrive at a solution. It's a powerful tool for problem-solving in all areas of life, whether it's figuring out how to budget your finances, planning a project, or making a difficult decision.

Moreover, algebraic thinking fosters critical thinking skills. When you solve an algebraic problem, you're not just memorizing formulas and procedures; you're actively engaging with the problem. You're analyzing the information, identifying the key variables, and developing a strategy for finding the solution. This process strengthens your critical thinking skills, which are essential for success in any field.

Furthermore, algebraic thinking promotes abstract reasoning. Algebra deals with symbols and variables, which represent abstract concepts. By working with these symbols, you're developing your ability to think abstractly, which is a valuable skill in many areas of life. Abstract reasoning allows you to see beyond the concrete details and identify the underlying principles and patterns.

In conclusion, mastering algebraic concepts equips you with the critical algebraic thinking skills needed to solve not just math problems, but also to approach challenges in various aspects of life. Keep honing these skills, guys, and you'll become confident problem-solvers in no time!

So, the answer to our original problem is 9. If you double 9 and add 5, you get 23. We've successfully solved the mystery! Keep practicing your algebra, guys, and you'll be amazed at what you can accomplish. Remember, every problem is just a puzzle waiting to be solved, and with the right tools and the right mindset, you can conquer any challenge. Let’s keep exploring the fascinating world of math together!