Solving Equations: 5x + 8x = 145 Explained

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Solving Equations: 5x + 8x = 145 Explained

Hey guys! Let's dive into solving the equation 5x + 8x = 145. This is a classic algebra problem, and we'll break it down step by step to make it super clear. Whether you're a math whiz or just starting out, this guide will help you understand how to solve this type of equation. We'll explore the basics of combining like terms and isolating the variable, which are fundamental concepts in algebra. Get ready to flex those math muscles! This equation involves combining like terms and solving for the unknown variable, 'x'. The goal is to find the value of 'x' that makes the equation true. Let's get started. Understanding this type of problem is crucial for building a strong foundation in algebra. These skills are not only useful in math class but also have applications in various real-world scenarios. Many students find this part of algebra tricky at first, but with a clear understanding of the steps involved, it becomes much easier. We will explore each step in detail so you can understand the methods. The best thing is that once you understand these basics you can tackle more complex equations with ease. So, buckle up and let's unravel this mathematical mystery together! We're going to break down the whole process, so don't sweat it if algebra isn't your favorite subject. This will also help you to brush up on basic algebra skills. We'll go through the fundamentals, making sure every step is crystal clear. By the end, you'll be able to solve this type of equation confidently. It is not something difficult; all you need is patience and good understanding. This equation solving is a stepping stone to more advanced topics. Knowing how to solve these problems is useful for any job or career you decide to have. The knowledge of equations can be applied to many situations that we encounter daily. Let's make it easy to understand the math, no pressure.

Combining Like Terms

Alright, first things first! When we look at 5x + 8x = 145, the initial step is to combine the like terms on the left side of the equation. Like terms are those that have the same variable raised to the same power. In our case, both '5x' and '8x' are like terms because they both have 'x' to the power of 1. To combine these, we simply add their coefficients (the numbers in front of the 'x'). So, 5 + 8 equals 13. Therefore, 5x + 8x simplifies to 13x. This is like saying if you have 5 apples and then get 8 more apples, you now have 13 apples total. The same principle applies to our equation, and combining these terms simplifies the equation, making it easier to solve. Understanding how to combine like terms is a fundamental skill in algebra. Once you master it, more complex equations are easy to solve. It reduces the complexity and makes the equation simpler. Remember this: Combine like terms by adding their coefficients. This helps you to solve equations more efficiently. When combining like terms, only the coefficients are added or subtracted. The variable and its exponent remain unchanged. For instance, in our equation, after combining the like terms, the variable 'x' remains unchanged. So, with 5x + 8x, we will have 13x. It's important to keep track of both the numbers and the variables. Remember, like terms are those with the same variable raised to the same power. Don't let those exponents confuse you! The goal is to make things manageable, one step at a time. This simple step will help us find the value of x.

Step-by-step example

Let's break down the combining like terms process in a more visual way:

  1. Identify Like Terms: In the equation 5x + 8x = 145, '5x' and '8x' are like terms.
  2. Add the Coefficients: Add the numbers (coefficients) in front of 'x': 5 + 8 = 13.
  3. Rewrite the Equation: Replace 5x + 8x with 13x. Now the equation becomes 13x = 145. Simple, right? See, combining like terms is not that hard. This makes it easier to work through the rest of the problem. This is a crucial step towards finding the value of 'x'. So, don't rush, and make sure you do it right. Once you combine your like terms, you're one step closer to solving the equation. The equation will look much simpler, and you'll be able to solve it more easily. Understanding how to combine these terms is fundamental to simplifying equations.

Isolating the Variable

Now that we've combined the like terms and simplified our equation to 13x = 145, the next goal is to isolate the variable 'x'. This means we want to get 'x' all by itself on one side of the equation. To do this, we need to get rid of the 13 that's multiplying 'x'. The method we use is to perform the opposite operation. Since 13 is multiplying 'x', we will divide both sides of the equation by 13. Remember, whatever you do to one side of the equation, you must do to the other side to keep it balanced. Isolating a variable is a fundamental concept in solving algebraic equations. It helps us find the specific value of an unknown variable that makes an equation true. This is the core of solving the equation. Once the variable is isolated, the solution becomes very apparent. To get 'x' alone, divide both sides of the equation by the coefficient. The main point is to make sure you keep the equation balanced. If you do something on one side, do it on the other side as well. This will ensure that we find the value of 'x' that makes the original equation true. The goal is to get 'x' by itself, and with this knowledge, you are ready to move on. This process ensures that the equation remains equivalent. Let's make it a little easier.

Step-by-step example

Let's break down the process of isolating the variable:

  1. Original Equation: 13x = 145
  2. Divide Both Sides by 13: To isolate 'x', divide both sides of the equation by 13: (13x) / 13 = 145 / 13.
  3. Simplify: On the left side, the 13s cancel out, leaving 'x'. On the right side, 145 divided by 13 equals 11.15 (approximately).
  4. Final Equation: x = 11.15. Now, 'x' is isolated, and we have solved for its value. See? Easy-peasy! This step directly gives us the value of 'x'. The ability to isolate variables is a crucial skill in algebra. The method is the same in all equations. Understanding these steps allows you to approach a wide variety of algebra problems with confidence. It is a fundamental concept in solving algebra equations, and it can be applied in many situations.

Solving for x and Understanding the Solution

After dividing both sides of the equation by 13, we found that x ≈ 11.15. This value of 'x' is the solution to our equation 5x + 8x = 145. What this means is that if you substitute 11.15 back into the original equation for 'x', the equation should be approximately true. In other words, when you plug 11.15 into the equation, both sides should be equal or very close due to the rounding. In this case, 5 * 11.15 + 8 * 11.15 = 145. This confirms that our solution is correct. To be sure that our solution is correct, we can also verify the answer by plugging it into the original equation. This is a critical step in problem-solving. It's always a great practice to check your answers. It's a key part of your mathematical journey. This helps you to have more confidence in your answers. Practice these concepts, and you will become more comfortable with algebraic equations. Understanding that the solution satisfies the original equation is crucial. This step is about verifying your findings and making sure everything makes sense. Remember, solving an equation means finding the value of the variable that makes the equation true. Knowing how to solve for 'x' opens doors to understanding many more complex equations. The more you work with it, the more familiar it will become. Let's make sure that you are confident with your problem-solving capabilities.

Checking Your Work

To make sure our answer is correct, let's substitute 'x' with 11.15 in the original equation, 5x + 8x = 145:

  1. Substitute x: 5 * 11.15 + 8 * 11.15 = 145
  2. Calculate: 55.75 + 89.2 = 144.95
  3. Check: The result is approximately 145. This confirms that our solution, x ≈ 11.15, is correct. This is the essence of solving for 'x'. Checking your work reinforces your understanding. See? Simple and correct. The more you do it, the better you'll get. Always remember to check your work; it's a great habit. Good job, you solved the equation!

Conclusion: Mastering the Equation

So, there you have it! We've successfully solved the equation 5x + 8x = 145. We combined like terms, isolated the variable 'x', and found its value. These steps are fundamental to algebra, and mastering them is crucial for your future math studies. Solving equations like these builds a solid foundation for more advanced topics in mathematics. Remember, practice is key. The more you work through these types of problems, the more confident you'll become. Keep practicing, and you'll become a pro at solving equations in no time! The methods we've used can be applied to many different types of algebra problems. Keep at it, and you'll see your understanding grow. Now that you've worked through this problem, you're one step closer to math mastery! Go out there and conquer those equations, guys! Feel free to ask if you have any questions. You did a great job following along. Remember, every step is important, and practice is the key to success. You've got this!