Solving Linear Equations: 10x + 25(x - 3) = 275
Let's break down how to solve the linear equation 10x + 25(x - 3) = 275. If you're scratching your head thinking about algebra, don't sweat it! We'll go through each step together, making it super easy to understand. We'll start by simplifying the equation, then isolate 'x' to find its value. By the end of this, you'll not only solve this specific equation but also gain a solid understanding of how to tackle similar problems. Ready? Let's dive in!
Understanding the Equation
The equation we're tackling is 10x + 25(x - 3) = 275. This is a linear equation, meaning the highest power of our variable 'x' is 1. To solve it, our goal is to isolate 'x' on one side of the equation. This involves a few key steps: distributing, combining like terms, and using inverse operations. Think of it like peeling an onion – each layer gets us closer to the core (which is finding the value of 'x'). Understanding the structure of the equation is the first step in making it less intimidating and more manageable. Remember, every equation is just a puzzle waiting to be solved, and we've got the tools to crack it!
Step-by-Step Solution
1. Distribute the 25
The first thing we need to do is get rid of those parentheses. We do this by distributing the 25 across the (x - 3) term. This means we multiply 25 by both 'x' and '-3'.
25 * x = 25x 25 * -3 = -75
So, our equation now looks like this: 10x + 25x - 75 = 275. See? We're already making progress!
2. Combine Like Terms
Next up, let's simplify by combining the 'x' terms on the left side of the equation. We have 10x and 25x. Adding these together gives us:
10x + 25x = 35x
Now, our equation is even simpler: 35x - 75 = 275. We're tidying things up nicely!
3. Isolate the Variable Term
We want to get the term with 'x' (which is 35x) by itself on one side of the equation. To do this, we need to get rid of the '-75'. We do this by adding 75 to both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced.
35x - 75 + 75 = 275 + 75 35x = 350
Looking good! Now we're just one step away from finding 'x'.
4. Solve for x
Finally, to isolate 'x', we need to get rid of the 35 that's multiplying it. We do this by dividing both sides of the equation by 35.
35x / 35 = 350 / 35 x = 10
And there you have it! The solution to the equation 10x + 25(x - 3) = 275 is x = 10.
Verification
To make sure we didn't make any mistakes along the way, let's plug our solution (x = 10) back into the original equation and see if it holds true.
Original equation: 10x + 25(x - 3) = 275
Substitute x = 10:
10(10) + 25(10 - 3) = 275 100 + 25(7) = 275 100 + 175 = 275 275 = 275
It checks out! Both sides of the equation are equal, so we know that x = 10 is indeed the correct solution. Always verify your solutions – it's a great way to catch any errors and build confidence in your algebra skills.
Alternative Approaches
While we solved this equation using a specific step-by-step method, there are often multiple ways to arrive at the same answer. For instance, you could choose to distribute and simplify in a slightly different order, or you might prefer to use different algebraic manipulations. The key is to understand the underlying principles and apply them in a way that makes sense to you. Don't be afraid to experiment and find what works best for your learning style. The more you practice, the more comfortable you'll become with different approaches, and the quicker you'll be able to solve equations efficiently.
Common Mistakes to Avoid
When solving equations like this, there are a few common pitfalls to watch out for. One frequent mistake is forgetting to distribute properly. Make sure you multiply the number outside the parentheses by every term inside. Another common error is combining unlike terms – you can only add or subtract terms that have the same variable and exponent. Also, be careful with your signs (positive and negative) when adding, subtracting, multiplying, or dividing. Keeping track of these details will help you avoid mistakes and ensure you get the correct solution. Pay close attention to detail, and always double-check your work!
Practice Problems
Now that you've seen how to solve this equation, let's put your skills to the test with a few practice problems. Remember, the more you practice, the better you'll become at solving algebraic equations. Here are a couple to get you started:
- 5x + 12(x - 2) = 64
- 8(y + 3) - 2y = 42
Work through these problems step-by-step, just like we did in the example. Don't be afraid to make mistakes – that's how we learn! And if you get stuck, go back and review the steps we covered earlier. With a little practice, you'll be solving equations like a pro in no time!
Conclusion
So, to recap, we successfully solved the equation 10x + 25(x - 3) = 275 and found that x = 10. We did this by distributing, combining like terms, isolating the variable term, and finally solving for 'x'. Remember to always verify your solution to ensure accuracy. With a solid understanding of these steps and a bit of practice, you can confidently tackle similar linear equations. Keep practicing, and you'll be amazed at how quickly your algebra skills improve! You got this! Algebraic equations might seem daunting at first, but with a systematic approach and consistent practice, anyone can master them. Keep breaking down complex problems into smaller, manageable steps, and don't be afraid to ask for help when you need it. Solving equations is like building a muscle – the more you exercise it, the stronger it gets. So keep challenging yourself, and watch your problem-solving abilities soar! And remember, math can actually be pretty fun once you get the hang of it!