Solve: Triple A Number Minus 8 Equals Double Plus 10

Nov 3, 2025 by SLV Team 53 views

Hey guys! Let's dive into this fun math problem where we're trying to figure out a mystery number. It sounds a bit like a word puzzle, right? We're given a scenario: "The triple of a number, decreased by 8, is equal to double the number increased by 10." Our mission, should we choose to accept it, is to crack the code and find out what that number is. Don't worry, we'll break it down step by step, making it super easy to understand. Math can be like detective work, and we're the detectives! So, grab your thinking caps, and let's get started on this mathematical adventure!

Understanding the Problem

Before we start crunching numbers, let's make sure we really understand what the problem is asking. The key is to translate the words into a mathematical equation. This might sound intimidating, but it's like learning a new language – once you get the basics, it becomes much easier. Our main goal here is to convert the sentence into something we can actually work with. We need to identify the unknowns and the relationships between them. Think of it as setting up the pieces on a chessboard before the game begins. If we get this initial setup right, the rest of the solution will flow much more smoothly. So, let’s take our time and make sure we're on the same page before we start solving!

Breaking Down the Sentence: The problem states: "The triple of a number, decreased by 8, is equal to double the number increased by 10." Let's dissect this piece by piece.
Identifying the Unknown: The most crucial part is figuring out what we're trying to find. In this case, it's "a number." Since we don't know what this number is yet, we'll call it something generic, like 'x'. This 'x' is our mystery variable, the thing we're trying to solve for.
Translating into Math: Now, let’s turn those words into mathematical expressions:
- "The triple of a number" means 3 times our number, which we call 'x'. So, this becomes 3x.
- "Decreased by 8" tells us we need to subtract 8 from our previous expression. So, we now have 3x - 8.
- "Is equal to" is a straightforward one – it just means an equals sign (=).
- "Double the number" translates to 2 times our number 'x', or 2x.
- "Increased by 10" means we add 10 to the double of the number. So, we have 2x + 10.
Forming the Equation: Now, we can put it all together. The left side of our equation will be the triple of the number decreased by 8 (3x - 8), and the right side will be double the number increased by 10 (2x + 10). Joining these with the "is equal to" sign gives us our complete equation: 3x - 8 = 2x + 10.

By carefully breaking down the sentence and translating each part into mathematical terms, we've successfully transformed a word problem into a solvable equation. This is a huge step, guys! Now we're ready to roll up our sleeves and actually solve for 'x'.

Setting Up the Equation

Alright, now that we've decoded the word problem, we've landed on our equation: 3x - 8 = 2x + 10. This equation is like the blueprint we'll use to find our mystery number. Think of it as a balancing scale – whatever we do to one side, we need to do to the other to keep it balanced. This principle is crucial for solving equations correctly. Our ultimate goal is to isolate 'x' on one side of the equation, so we know exactly what it equals. This might involve moving terms around, adding, subtracting, multiplying, or dividing. But don't worry, we'll take it one step at a time. The key is to be systematic and keep the equation balanced throughout the process. So, let's get this equation ready for solving!

The Goal: Isolate 'x': Remember, our main aim is to get 'x' all by itself on one side of the equation. This means we need to move everything else – the numbers and the other 'x' term – to the other side. It’s like clearing a path so 'x' can reveal its true value.
Moving Terms Around: To isolate 'x', we need to move the 2x term from the right side to the left side. We can do this by subtracting 2x from both sides of the equation. Why subtract? Because by subtracting 2x from 2x, we effectively cancel it out on the right side. But remember, we have to do the same thing on the left side to keep the equation balanced. This gives us:
```
3x - 8 - 2x = 2x + 10 - 2x
```
Simplifying this, we get:
```
x - 8 = 10
```
See how the 2x has disappeared from the right side? We're one step closer!
Next Steps: Now we have x - 8 = 10. We still need to get 'x' completely alone. The -8 is standing in our way. To get rid of it, we'll do the opposite operation – we'll add 8 to both sides of the equation. This is another example of keeping the equation balanced.

By carefully moving terms around while maintaining the balance of the equation, we’re steadily marching towards our solution. We’ve already simplified the equation quite a bit, and now we're just one step away from finding out what 'x' really is!

Solving for the Unknown

Okay, we've arrived at the final stretch! We've simplified our equation to x - 8 = 10. Now, it's time to get 'x' completely on its own and reveal its value. Remember how we talked about keeping the equation balanced? This is where that principle really shines. To isolate 'x', we need to get rid of that -8. The way we do that is by doing the opposite operation – adding 8. But, and this is crucial, we need to add 8 to both sides of the equation to keep things balanced. Think of it as adding the same weight to both sides of a scale so it stays level. Once we do this final step, the mystery of 'x' will be solved!

Adding to Both Sides: As we discussed, we need to add 8 to both sides of the equation to cancel out the -8 on the left side. This looks like this:
```
x - 8 + 8 = 10 + 8
```
Simplifying the Equation: Now, let's simplify. On the left side, -8 + 8 cancels out, leaving us with just 'x'. On the right side, 10 + 8 equals 18. So, our equation now reads:
```
x = 18
```
Boom! That's it! We've found our number. 'x' equals 18. It's like we've unlocked a secret code. All the hard work of translating the words, setting up the equation, and carefully moving terms around has paid off.

By performing this final step of adding to both sides and simplifying, we've successfully isolated 'x' and discovered its value. This is a huge accomplishment, guys! Now that we have our solution, there's just one thing left to do: check our work to make sure we got it right.

Checking the Solution

We've solved for 'x', and we think it's 18. That's awesome! But before we do a victory dance, it's super important to make sure our answer is correct. Think of it as proofreading your work before you hand it in. We want to be absolutely sure that 18 fits the original problem. This is where checking our solution comes in. We're going to take our answer, plug it back into the original equation, and see if both sides balance out. If they do, we know we've cracked the code! If they don't, it means we need to go back and see where we might have made a mistake. So, let’s put our solution to the test!

Going Back to the Original Equation: Remember our original equation? It was 3x - 8 = 2x + 10. This is the equation we formed directly from the word problem, so it's the ultimate test of whether our solution works.
Plugging in Our Solution: Now, we're going to replace 'x' in the equation with our answer, which is 18. This gives us:
```
3(18) - 8 = 2(18) + 10
```
Simplifying Both Sides: Let's simplify each side of the equation separately:
- Left side: 3(18) = 54, so we have 54 - 8, which equals 46.
- Right side: 2(18) = 36, so we have 36 + 10, which also equals 46.
Checking for Balance: Look at that! Both sides of the equation simplify to 46. This means our equation balances perfectly: 46 = 46. This is exactly what we want to see!

By plugging our solution back into the original equation and verifying that both sides balance, we've confirmed that our answer is correct. This is a huge confidence booster, guys! We can now say with certainty that 'x' truly equals 18.

Final Answer and Recap

Alright, we've reached the finish line! After carefully translating the word problem, setting up the equation, solving for 'x', and checking our solution, we've definitively found the answer. It's like we've completed a mathematical quest, and we've emerged victorious! So, let's shout it from the rooftops (or, you know, just state it clearly): the number we were looking for, the 'x' in our equation, is 18. This means that when you triple 18 and subtract 8, you get the same result as when you double 18 and add 10. Pretty neat, huh?

The Final Answer: The solution to the problem "The triple of a number, decreased by 8, is equal to double the number increased by 10" is:
```
x = 18
```
That's it! We've nailed it.
Quick Recap of Steps: Let's quickly recap the journey we took to get here. This is a great way to reinforce what we've learned:
1. Understanding the Problem: We carefully read and dissected the word problem, identifying the unknown and the relationships between the numbers.
2. Setting Up the Equation: We translated the words into a mathematical equation, representing the unknown number with the variable 'x'.
3. Solving for the Unknown: We used algebraic principles to isolate 'x' on one side of the equation, step by step, keeping the equation balanced throughout.
4. Checking the Solution: We plugged our solution back into the original equation to verify that it worked, ensuring that both sides balanced.

By stating the final answer clearly and recapping the steps we took, we've not only solved the problem but also solidified our understanding of the process. This is what math is all about, guys! It's about taking a challenge, breaking it down, and systematically working towards a solution. We did it! Great job, everyone!

In conclusion, the key to solving word problems like this is to break them down into smaller, manageable parts. Translate the words into mathematical expressions, set up an equation, solve for the unknown, and always check your solution. With practice, you'll become a word problem-solving pro! Keep up the great work, and remember, math can be fun!