Solving The Equation: 4x - 34 + 62 Step-by-Step

Hey guys! Let's dive into how to solve the equation 4x - 34 + 62. This is a pretty straightforward problem in algebra, and we'll break it down into easy-to-follow steps. Our goal is to isolate 'x' and find its value. Ready? Let's get started!

Understanding the Basics of Algebraic Equations

Before we jump into the specific problem, let's refresh our understanding of basic algebraic equations. In algebra, we often deal with equations that involve variables (like 'x'), constants (numbers like -34 and 62), and mathematical operations (+, -, ×, ÷). The core concept is that an equation represents a balance. The left side of the equation is equal to the right side. Our main aim when solving an equation is to find the value of the variable that makes the equation true. This is done by manipulating the equation using mathematical rules to isolate the variable on one side. Remember, whatever you do to one side of the equation, you must do to the other side to keep things balanced. It's like a seesaw; to keep it level, you need to add or remove weight from both sides equally. Also, the key to tackling these problems is patience and attention to detail. Always double-check your calculations and ensure you're applying the operations correctly. We'll be using the following principles throughout this process: Combining Like Terms, the addition/subtraction property of equality, and the multiplication/division property of equality. Keep in mind that the goal is to simplify the equation step by step until we have something that looks like 'x = a number'. The skills learned here form a crucial foundation for more complex mathematical concepts, so understanding each step is super important. In essence, solving an algebraic equation involves a series of strategic moves. These moves are guided by the principles of maintaining equality while working towards isolating the variable. Understanding the rules and practicing these techniques will help you develop confidence and proficiency in solving equations. Let's move forward with our equation and see how these concepts play out in practice.

Combining Like Terms

In our equation 4x - 34 + 62, we need to simplify by combining like terms. In this equation, the numbers -34 and 62 are constants, meaning they don't have any variables attached to them. That makes them like terms, so we can add them together directly. Let's do the math:

• 62 - 34 = 28

So, when we combine the like terms, our equation becomes:

• 4x + 28

This simplified form is the first step toward isolating 'x'. Essentially, combining like terms is about making the equation cleaner and easier to work with. By reducing multiple terms to a single term, we simplify the equation. Keep in mind that you can only combine like terms. For example, you can't combine a term with 'x' (like 4x) with a constant number (like 28). The process of simplifying the equation is just making it easier to solve. The idea is to make the equation as simple as possible, making each step clearer and less prone to errors. This step lays the groundwork for the next step. Be careful with the signs (positive and negative) when you're combining the terms. A small mistake in this step can affect your final answer.

Isolating the Variable

Now, our equation is 4x + 28 = something (we'll figure out the 'something' later, or you can assume it equals 0). Our goal is to get 'x' by itself. To do this, we need to get rid of the +28. We use the property of equality, which says that if you perform the same operation on both sides of the equation, the equation remains balanced. So, to eliminate the +28, we subtract 28 from both sides:

• 4x + 28 - 28 = 0 - 28

This simplifies to:

• 4x = -28

We're getting closer! By subtracting 28 from both sides, we've isolated the term with 'x' on one side. This is a critical step in solving any linear equation. The basic idea is to systematically eliminate everything that is not 'x' from the side where 'x' is located. Each step brings us closer to the ultimate solution. Don't forget to do the operation on both sides. Otherwise, your equation will be unbalanced, leading to an incorrect answer. This concept is fundamental and forms the backbone of solving linear equations. If you're comfortable with this concept, you can move on to more advanced problems. Remember, practice makes perfect! The more you do these problems, the faster and more confident you will become.

Solving for x

We now have 4x = -28. Here, 'x' is multiplied by 4. To isolate 'x', we need to do the opposite operation: division. We'll divide both sides of the equation by 4:

• 4x / 4 = -28 / 4

This simplifies to:

• x = -7

And there we have it! We've solved for 'x'. The value of x that satisfies the original equation is -7. This step is about isolating the variable, which means getting the variable on its own on one side of the equation. The goal is to make it simple enough that we can see the value of 'x' directly. Be careful when dividing; pay attention to the signs. Remember that a negative number divided by a positive number results in a negative number. This might be a good time to double-check your calculations and make sure everything is in order. With this last step, you've successfully solved for 'x'. Keep up the good work. Remember, practice is the key to mastering these types of problems.

Checking Your Work

It's always a good idea to check your work. To do this, substitute the value you found for 'x' back into the original equation and see if it holds true. Our original equation was 4x - 34 + 62 = 0 (let's assume, it simplifies to 0 in the actual problem setting). We found that x = -7. Let's plug this back in:

• 4(-7) - 34 + 62

Calculate it step by step:

• -28 - 34 + 62

• -62 + 62

• 0

Since the equation holds true (equals 0), our solution x = -7 is correct! Checking your answers is not just about verifying the result. It's an important skill that strengthens your understanding of the problem. It can also help you spot any mistakes you've made during the process. By substituting your solution back into the original equation, you can verify if your calculation makes sense. Always make it a habit to double-check your solution. Even if you're pressed for time, a quick check can save you from a lot of frustration down the line. Being a careful worker, in general, will benefit you in the long run.

Tips for Solving Similar Equations

Here are a few tips that can help you solve similar equations more easily:

• Always simplify first: Combine like terms at the start. This makes the equation less complex and easier to manage.
• Keep it balanced: Remember to perform the same operations on both sides of the equation.
• Double-check signs: Pay close attention to positive and negative signs; a small mistake can change your answer.
• Practice regularly: The more you practice, the better you'll become at solving these problems. Try different types of equations to improve your skills.
• Break it down: If an equation looks complicated, break it down into smaller steps. This will make the solving process easier and reduce the chance of errors.
• Don't give up: If you get stuck, don't give up. Review your steps, and try again. You'll get it!

By following these tips, you'll be well on your way to becoming a pro at solving algebraic equations. Remember, practice is the key! Keep at it, and you'll soon find these equations easy to solve. These are some important tips. The more you know, the more you'll succeed. So, get out there, do some more equations, and keep learning.

Conclusion

Solving the equation 4x - 34 + 62 involves combining like terms, isolating the variable, and solving for 'x'. We found that x = -7. Remember to check your answer to ensure it's correct. Keep practicing, and you'll build your skills and confidence in algebra. The process of solving algebraic equations might seem challenging at first. But with practice and patience, you will gain confidence and expertise. Keep practicing! You are on your way to a new level. Always remember that a strong foundation in algebra can open up doors to more advanced mathematical concepts. Keep up the good work, and keep on learning!