Solving The Algebraic Equation 7x + 43 + 6x + 15 Step-by-Step Guide
Hey guys! Ever stumbled upon an algebraic equation that looks like a jumbled mess of numbers and letters? Don't worry, it happens to the best of us! Today, we're going to break down a common type of equation and show you exactly how to solve it. We'll take the equation 7x + 43 + 6x + 15 = ? and turn it from something intimidating into a piece of cake. So, grab your pencils, and let's dive in!
Understanding the Basics of Algebraic Equations
Before we jump into solving this specific equation, let's quickly brush up on some fundamental concepts. In algebra, we often use letters, most commonly 'x,' to represent unknown values. These letters are called variables. The goal of solving an equation is to figure out the value of this variable that makes the equation true. Think of it like a puzzle where you need to find the missing piece.
An equation is like a balanced scale. Both sides of the equals sign (=) must have the same value. To keep the scale balanced, whatever operation you perform on one side, you must also perform on the other. This is a crucial principle to remember when solving equations.
Terms are the individual components of an equation, separated by plus (+) or minus (-) signs. In our equation 7x + 43 + 6x + 15, the terms are 7x, 43, 6x, and 15. Like terms are terms that have the same variable raised to the same power. In our example, 7x and 6x are like terms, and 43 and 15 are like terms (they are both constants).
Combining like terms is a key step in simplifying equations. It involves adding or subtracting the coefficients (the numbers in front of the variables) of like terms. For instance, 7x + 6x can be combined into 13x. This makes the equation simpler and easier to work with. Remember, you can only combine like terms; you can't combine a term with 'x' with a constant term.
Understanding these basics will make the process of solving equations much smoother and less confusing. It's like having the right tools before you start a DIY project; it makes the whole task much more manageable and enjoyable. So, with these concepts in mind, let's get back to our equation and see how we can apply them.
Step-by-Step Solution: 7x + 43 + 6x + 15 = ?
Okay, let's tackle the equation 7x + 43 + 6x + 15 = ? step by step. We're going to break it down into manageable chunks so you can see exactly how it works. Trust me, it's not as scary as it looks!
Step 1: Combine Like Terms
The first thing we want to do is simplify the equation by combining like terms. Remember, like terms are those that have the same variable (in this case, 'x') or are constants (just numbers). So, let's group them together:
- We have
7xand6x. These are like terms because they both have 'x'. - We also have
43and15. These are like terms because they are both constants.
Now, let's add the like terms together:
7x + 6x = 13x43 + 15 = 58
So, after combining like terms, our equation now looks like this: 13x + 58 = ?. See? We've already made it simpler!
Step 2: Isolate the Variable Term
Our next goal is to isolate the term with the variable (in this case, 13x) on one side of the equation. This means we need to get rid of the 58 that's being added to it. Remember our balanced scale analogy? Whatever we do to one side, we must do to the other.
To get rid of the 58, we need to subtract it from both sides of the equation:
13x + 58 - 58 = 0 - 58
This simplifies to:
13x = -58
Great! Now we have the variable term isolated on one side. We're getting closer!
Step 3: Solve for x
The final step is to solve for 'x'. Right now, we have 13x, which means 13 times 'x'. To find the value of 'x', we need to undo this multiplication. How do we do that? By dividing! We'll divide both sides of the equation by 13:
13x / 13 = -58 / 13
This simplifies to:
x = -58 / 13
Now, you can leave the answer as a fraction, -58/13, or you can convert it to a decimal. If you divide -58 by 13 using a calculator, you'll get approximately:
x ≈ -4.46
And there you have it! We've solved the equation. The value of 'x' that makes the equation true is approximately -4.46. High five!
Common Mistakes to Avoid When Solving Equations
Solving equations can be tricky, and it's easy to make mistakes if you're not careful. But don't worry, we're here to help you avoid those pitfalls! Here are some common errors to watch out for:
Forgetting to Distribute
Sometimes, equations will have parentheses, like this: 2(x + 3) = 10. Before you can start combining like terms, you need to distribute the number outside the parentheses to each term inside. This means multiplying the 2 by both the 'x' and the '3'. A common mistake is to only multiply by the first term, but you need to multiply by all of them!
Incorrectly Combining Like Terms
Remember, you can only combine terms that are alike. You can't add a term with 'x' to a constant term. For example, in the expression 3x + 5 + 2x, you can combine 3x and 2x to get 5x, but you can't combine the 5x with the 5. It's like trying to add apples and oranges – they're just not the same!
Not Performing the Same Operation on Both Sides
This is a big one! The golden rule of solving equations is that whatever you do to one side, you must do to the other. If you add 5 to the left side, you need to add 5 to the right side. If you divide the right side by 2, you need to divide the left side by 2. If you forget this, your equation will become unbalanced, and you won't get the correct answer.
Sign Errors
Sign errors are super common, especially when dealing with negative numbers. Pay close attention to whether a term is positive or negative, and make sure you're applying the correct operations. For example, subtracting a negative number is the same as adding a positive number, so - (-3) becomes + 3. Double-check your signs to avoid these sneaky mistakes!
Rushing Through the Steps
It's tempting to rush through the steps, especially if you feel confident, but this is where mistakes often happen. Take your time, write out each step clearly, and double-check your work as you go. It's better to be slow and accurate than fast and wrong!
By being aware of these common mistakes, you can avoid them and become a much more confident and accurate equation solver. Remember, practice makes perfect, so keep working at it, and you'll get there!
Practice Problems: Test Your Skills!
Alright, guys, now it's your turn to put those equation-solving skills to the test! Practice is key to mastering any new concept, and solving equations is no different. So, let's dive into some practice problems to solidify your understanding. Grab a pencil and paper, and let's get started!
Here are a few equations for you to try:
5x + 12 + 2x - 3 = 313(x - 2) + 4x = 298x - 15 = 5x + 94(2x + 1) = 3(x - 2) + 1510 - 2x + 5x = 4x - 6
Take your time and work through each problem step by step. Remember to combine like terms, isolate the variable, and solve for 'x'. Don't forget to double-check your work to avoid those common mistakes we talked about earlier!
If you get stuck, don't worry! Go back and review the steps we covered in this guide. Pay attention to how we combined like terms, isolated the variable, and performed operations on both sides of the equation. Sometimes, just seeing the process again can help you figure out where you went wrong.
After you've attempted the problems, it's a great idea to check your answers. You can use an online equation solver or ask a friend or teacher to look over your work. The most important thing is to understand the process and learn from any mistakes you make.
Practice problems are like training exercises for your brain. The more you practice, the stronger your equation-solving muscles will become. So, keep at it, and you'll be tackling even the trickiest equations with confidence in no time!
Conclusion: You're an Equation-Solving Rockstar!
Wow, you've come a long way! We've journeyed through the world of algebraic equations, breaking down the equation 7x + 43 + 6x + 15 = ? step by step. You've learned how to combine like terms, isolate variables, and solve for 'x'. You've also discovered common mistakes to avoid and had a chance to practice your skills with some challenging problems. Give yourself a pat on the back – you've earned it!
Solving equations is a fundamental skill in mathematics, and it's one that you'll use again and again in various contexts. Whether you're studying more advanced math topics, working on science problems, or even managing your finances, the ability to solve equations will come in handy.
Remember, the key to success in math is practice and persistence. Don't get discouraged if you don't understand something right away. Keep working at it, ask questions, and seek out help when you need it. The more you practice, the more confident and skilled you'll become.
So, go forth and conquer those equations! You now have the tools and knowledge to tackle a wide range of algebraic problems. Keep practicing, keep learning, and keep pushing yourself to grow. You've got this! And remember, math can be fun – especially when you know what you're doing. Keep that equation-solving rockstar attitude, and you'll shine in all your mathematical endeavors!
