Unraveling The Equation: Finding 'a' Step-by-Step

Hey guys! Let's dive into a fun math problem today. We're going to crack the code and find the value of 'a' in the equation: a5a - a37 + 4a = 23. Sounds exciting, right? Don't worry, it's not as scary as it looks. We'll break it down step by step, making it super easy to understand. So, grab your pencils and let's get started on this mathematical adventure! Remember, understanding the process is key, and we'll make sure you get it!

Decoding the Equation: Initial Setup

Alright, let's start by looking at our equation again: a5a - a37 + 4a = 23. The first thing we need to do is to understand what each part of the equation means. In this case, 'a5a' is a number where 'a' is the hundredths digit, 5 is the tens digit and 'a' is the units digit, 'a37' is a number where 'a' is the hundreds digit, 3 is the tens digit and 7 is the units digit. Also we have a multiplication between the numbers, and the value we have to find. This kind of problem often appears in basic algebra, and solving it involves isolating the variable 'a' to find its numerical value. So, our primary goal is to rewrite the equation in a way that allows us to find the value of the unknown variable, 'a'. Think of it like a puzzle; we're arranging the pieces to solve for 'a'. This includes identifying and clarifying the different terms and ensuring everything is aligned to make the math calculations easier. Before we begin, let's also remember the basic rules of algebra: always perform the operations in the correct order (PEMDAS/BODMAS) and keep the equation balanced by doing the same thing to both sides. It's like a seesaw; to keep it balanced, any action done on one side must also be done on the other side of the equation. This is our foundation; now we are ready to move forward.

Now, let's focus on the initial organization. Before anything else, let's rewrite the equation in a more familiar format that is easier to read and understand. To rewrite a5a, we need to recognize that it represents a number. In this case, 'a' represents the hundreds digit, '5' the tens digit and 'a' the units digit. So we can write it as 100a + 50 + a = 101a + 50. The expression a37 represents a number where 'a' is the hundreds digit, 3 the tens digit, and 7 the units digit. We can write it as 100a + 30 + 7 = 100a + 37. Now our equation is looking like this: 101a + 50 - (100a + 37) + 4a = 23. This simple reformatting is just the first step in simplifying the equation. It makes it easier to keep track of the variables and avoid errors. Remember, paying close attention to these small details can significantly streamline your way through a problem. Next, we will simplify further, removing the parentheses and combining like terms. This organized approach to solving the equation is critical for accuracy and efficiency. Just by carefully laying out the foundation, we're making the rest of the problem-solving process significantly easier and less prone to errors. Ready for the next stage? Let's get to it!

Simplifying the Equation: Combining Terms

Now, we're going to tidy up the equation. Let's start by removing those parentheses. Remember, when there is a minus sign in front of a parenthesis, we have to change the sign of each term inside it. So, - (100a + 37) becomes -100a - 37. Our equation is now: 101a + 50 - 100a - 37 + 4a = 23. See, much cleaner already! Next step? Combine the 'a' terms. We have 101a, -100a, and +4a. Adding those up, we get (101 - 100 + 4)a = 5a. Then, let's combine the constants: 50 - 37 = 13. Putting it all together, our equation simplifies to: 5a + 13 = 23. Isn't that better? See how combining like terms has simplified things? We have moved from a complex equation to a simple one! It is like cleaning up your room, and everything looks more manageable. It really makes things so much easier. The goal of this step is to get all the 'a' terms on one side of the equation and all the numbers on the other side. This is all about gathering similar terms together. Doing this makes solving for 'a' significantly more straightforward. Remember, with each step, we are getting closer to finding the value of 'a'! Keep up the great work, everyone. We are doing great!

Let's reflect on what we've done so far. We started with a5a - a37 + 4a = 23. We re-wrote this with the proper values, getting: 101a + 50 - (100a + 37) + 4a = 23. Then, after removing the parentheses and combining like terms, we simplified it to 5a + 13 = 23. The reason for combining all the terms together is that it allows us to consolidate all the information to find our solution. By bringing the similar terms together, we're able to see the equation's true form more clearly. This process removes the clutter and makes it easier to work towards the final answer. Keep these steps in mind as they are crucial for solving algebraic equations. Remember, the goal is always to isolate the variable and find its value. So, now that we have a cleaner equation, we can continue to isolate 'a'. Next, we will continue simplifying this equation.

Isolating 'a': Finding the Value

Time to find the final value of 'a'! Now we have the simplified equation: 5a + 13 = 23. Our goal is to isolate 'a', which means getting 'a' by itself on one side of the equation. The first step is to get rid of that '+ 13'. To do this, we subtract 13 from both sides of the equation. This gives us: 5a + 13 - 13 = 23 - 13. Simplifying this, we get: 5a = 10. You see, by performing the same operation on both sides, we keep the equation balanced, and we are one step closer to our solution. Next up, we will deal with the '5' in front of 'a'. Since '5a' means '5 multiplied by a', to isolate 'a', we do the opposite: divide both sides of the equation by 5. Our equation now looks like this: 5a / 5 = 10 / 5. This simplifies to: a = 2. And there we have it! We have successfully found the value of 'a'. Congratulations, everyone! We have solved the equation. Wasn't that fun?

So, to recap, we isolated 'a' by first subtracting 13 from both sides, then dividing both sides by 5. Remember, when solving equations, we must always keep them balanced. Any operation performed on one side must be performed on the other side. Think of it as a balance scale. To keep the scale balanced, if you add or remove weight from one side, you must do the same to the other side. This approach ensures that we arrive at the correct answer. By understanding and consistently applying these steps, you can confidently solve any algebraic equation. Keep practicing, and you will become a master! This whole process reinforces the basic principles of algebraic manipulation and how to isolate a variable. In solving this, you’ve not only solved a single problem but have armed yourself with a powerful tool for tackling a variety of math challenges!

Conclusion: Wrapping It Up

We did it, guys! We successfully found that a = 2 in the equation a5a - a37 + 4a = 23. It may have seemed tricky initially, but by breaking it down into smaller, manageable steps, it became a piece of cake. Remember, the key is to understand each step, simplify the equation, and isolate the variable. We started with a complex expression and systematically worked our way through to a simple solution. We hope you enjoyed this journey and learned something new today. Keep practicing and exploring, and you'll find that math can be really fun and rewarding! Always remember, the process is more important than the solution, and understanding each step is what truly matters.

Let's summarize the key takeaways: We started with a complex equation and rewrote it, simplified it, and isolated the variable 'a'. By combining like terms and performing inverse operations on both sides of the equation, we were able to reveal the value of 'a'. The steps we took, like carefully simplifying and organizing the equation, are skills that are applicable far beyond this particular problem. Understanding these concepts will help you build a solid foundation in mathematics and prepare you to tackle more complex challenges. So, pat yourselves on the back, and keep up the great work! Always remember, math is about the journey. Each step of the way is an opportunity to learn, grow, and strengthen your problem-solving skills. So keep exploring, keep questioning, and never stop learning! With each equation solved, you are building confidence and honing skills that will benefit you in all areas of life. Keep pushing yourselves to learn more and don’t be afraid to try new things. Math is like any other skill; with practice and persistence, you'll become a pro!