Solve 7x - 3(x + 1) = 13: A Step-by-Step Solution
Hey everyone! Let's dive into solving this algebraic equation: 7x - 3(x + 1) = 13. If you're scratching your head wondering where to even begin, don't worry! I'm here to break it down for you step-by-step, making it super easy to understand. We'll go through each part of the equation, simplify it, and find the value of 'x'. So, grab your pencil and paper, and let's get started!
Understanding the Equation
First, let's take a good look at our equation: 7x - 3(x + 1) = 13. This might seem a bit intimidating at first, but trust me, it's totally manageable. The key here is to follow the order of operations, which you might remember as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). We'll tackle the parentheses first, then move on to the rest.
Dealing with Parentheses
The first thing we need to do is handle the term inside the parentheses: (x + 1). But wait, there's also a -3 right outside the parentheses! That means we need to distribute the -3 across both terms inside the parentheses. This is a crucial step, so let's take it slow and make sure we get it right.
When we distribute the -3, we're essentially multiplying it by each term inside the parentheses. So, -3 * x gives us -3x, and -3 * 1 gives us -3. Now our equation looks like this: 7x - 3x - 3 = 13. See? We've already made some progress!
Combining Like Terms
Now that we've dealt with the parentheses, it's time to simplify the equation further. We can do this by combining what we call "like terms." Like terms are terms that have the same variable raised to the same power. In our equation, we have two terms with 'x': 7x and -3x. We can combine these by simply adding their coefficients (the numbers in front of the 'x').
So, 7x - 3x equals 4x. Now our equation looks even simpler: 4x - 3 = 13. We're getting closer to solving for 'x'!
Isolating the Variable
Our next goal is to isolate 'x' on one side of the equation. This means we want to get 'x' all by itself on either the left or the right side. To do this, we need to get rid of the -3 that's hanging out on the left side with the 4x.
Remember, in algebra, we can do the same thing to both sides of the equation and keep it balanced. So, to get rid of the -3, we can add 3 to both sides. This gives us: 4x - 3 + 3 = 13 + 3. The -3 and +3 on the left side cancel each other out, leaving us with 4x = 16.
Solving for x
We're almost there! We now have 4x = 16. To finally solve for 'x', we need to get rid of the 4 that's multiplying it. Again, we can do the same thing to both sides of the equation. This time, we'll divide both sides by 4. So, 4x / 4 = 16 / 4.
This simplifies to x = 4. And that's it! We've solved the equation! 🎉
Final Answer
So, the solution to the equation 7x - 3(x + 1) = 13 is x = 4. Wasn't that fun? (Okay, maybe not fun fun, but definitely satisfying!). Let's recap the steps we took to get here:
- Distribute the -3: 7x - 3x - 3 = 13
- Combine like terms: 4x - 3 = 13
- Add 3 to both sides: 4x = 16
- Divide both sides by 4: x = 4
Common Mistakes to Avoid
Now that we've successfully solved the equation, let's quickly touch on some common mistakes people make when tackling problems like this. Avoiding these pitfalls will help you nail algebraic equations every time!
Forgetting to Distribute
One of the most frequent errors is forgetting to distribute the number outside the parentheses to both terms inside. In our case, it's crucial to multiply the -3 by both x and 1. If you only multiply by one term, you'll end up with the wrong answer. So, always double-check that you've distributed correctly!
Sign Errors
Another common mistake is messing up the signs. Remember, a negative times a positive is a negative, and a negative times a negative is a positive. Pay close attention to the signs when you're distributing and combining like terms. A small sign error can throw off the entire solution.
Incorrect Order of Operations
As we discussed earlier, the order of operations (PEMDAS) is key. Make sure you handle parentheses and exponents before multiplication and division, and addition and subtraction. Skipping a step or doing things in the wrong order will lead to incorrect results. Treat your order of operations like a recipe, and follow it closely.
Not Combining Like Terms Properly
Combining like terms is a crucial simplification step. Make sure you're only combining terms that have the same variable raised to the same power. You can't combine 4x with just a number, like 3, because they're not like terms. Double-check that you're only combining terms that belong together.
Skipping Steps
It might be tempting to try and do everything in your head to save time, but skipping steps can often lead to mistakes. Writing out each step clearly helps you keep track of your work and makes it easier to spot errors. It's always better to be thorough and accurate than to rush and make a mistake.
Not Checking Your Answer
Finally, one of the best ways to avoid errors is to check your answer! Once you've solved for 'x', plug it back into the original equation and see if it works. If both sides of the equation are equal, you know you've got the right answer. If not, go back and look for mistakes in your work.
Real-World Applications of Algebra
Now that you've mastered solving this equation, you might be wondering, "Okay, but when am I ever going to use this in real life?" Well, algebra isn't just a bunch of abstract symbols and equations. It's a powerful tool that helps us solve problems in many different areas of life. Let's look at some examples.
Personal Finance
Algebra is super useful when it comes to managing your money. For instance, if you're trying to figure out how much you need to save each month to reach a financial goal, you can use algebraic equations to calculate it. You can also use algebra to understand interest rates, loan payments, and investment returns. It's like having a secret weapon for making smart financial decisions!
Imagine you want to buy a new gadget that costs $500, and you plan to save for it over 6 months. If you already have $100 saved, you can use algebra to figure out how much more you need to save each month. Let's say 'x' is the amount you need to save monthly. The equation would look something like this: 100 + 6x = 500. Solving for 'x' will tell you how much you need to save each month.
Home Improvement
Planning a home improvement project? Algebra can help! Whether you're calculating the amount of paint you need for a room, figuring out the dimensions for a new piece of furniture, or estimating the cost of materials, algebraic concepts can come in handy. It's all about using math to make your DIY dreams a reality.
For example, let's say you want to build a rectangular garden bed, and you have 20 feet of fencing. If you want the length of the bed to be twice the width, you can use algebra to find the dimensions. Let 'w' be the width and '2w' be the length. The perimeter (total fencing) can be represented as: 2w + 2(2w) = 20. Solving for 'w' will give you the width, and then you can find the length.
Cooking and Baking
Believe it or not, algebra even plays a role in the kitchen! When you're scaling a recipe up or down, you're essentially using algebraic proportions. You need to adjust the amounts of each ingredient to maintain the right ratios. Algebra helps you do this accurately, so your dishes turn out perfectly every time.
Suppose you have a recipe that makes 6 cookies, but you want to make 15. If the recipe calls for 1 cup of flour, you can use algebra to find out how much flour you need for 15 cookies. You can set up a proportion: 1 cup / 6 cookies = x cups / 15 cookies. Solving for 'x' will give you the amount of flour you need.
Travel and Navigation
Planning a trip? Algebra can help you figure out distances, travel times, and fuel costs. You can use equations to calculate how long it will take to drive a certain distance, or how much gas you'll need for your journey. It's like having a personal travel assistant powered by math!
Let's say you're driving 300 miles, and your car gets 25 miles per gallon. If gas costs $3 per gallon, you can use algebra to estimate your fuel cost. First, find out how many gallons you'll need: 300 miles / 25 miles per gallon = 12 gallons. Then, multiply the number of gallons by the cost per gallon: 12 gallons * $3 per gallon = $36. So, your estimated fuel cost is $36.
Time Management
Even managing your daily schedule can involve algebra! If you have a set amount of time and a list of tasks to complete, you can use algebra to allocate your time effectively. You can estimate how long each task will take and then create an equation to make sure everything fits within your timeframe.
Imagine you have 3 hours (180 minutes) to complete 4 tasks. If Task A takes 45 minutes, Task B takes 30 minutes, and Task C takes 25 minutes, you can use algebra to find out how much time you have left for Task D. Let 'x' be the time for Task D. The equation would be: 45 + 30 + 25 + x = 180. Solving for 'x' will tell you how much time you have for Task D.
Conclusion
So, there you have it! We've not only solved the algebraic equation 7x - 3(x + 1) = 13, but we've also explored some real-world applications of algebra. From managing your finances to planning a road trip, algebra is a valuable tool that can help you make informed decisions and solve problems in many areas of your life. Keep practicing, and you'll become an algebra whiz in no time! Remember, the solution to the equation 7x - 3(x + 1) = 13 is x = 4.
