Solving Equations Step By Step 3q + 5 + 2q - 5 = 65

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Introduction

Hey guys! Today, we're going to break down a common type of problem you'll see in algebra: solving linear equations. Specifically, we'll be tackling the equation 3q + 5 + 2q - 5 = 65. Don't worry if it looks a bit intimidating at first. We'll go through it step by step, making sure everything is crystal clear. Our goal is not just to find the answer, but also to understand the process so you can solve similar problems with confidence. So, grab your pencils, and let's dive in!

Understanding the Basics of Linear Equations

Before we jump into solving our specific equation, let's quickly recap what a linear equation is and the basic principles we use to solve them. Think of an equation like a balanced scale. Both sides of the equals sign (=) must always remain balanced. Whatever operation you perform on one side, you must also perform on the other side to keep the equation true. This principle is the foundation of solving equations. We use inverse operations (addition and subtraction, multiplication and division) to isolate the variable (in our case, 'q') on one side of the equation, allowing us to find its value. For example, if we have q + 3 = 7, we subtract 3 from both sides to get q = 4. This keeps the equation balanced and helps us find the solution. In essence, solving linear equations is all about strategically manipulating the equation while maintaining balance until we reveal the value of the unknown variable. This concept is crucial, and we'll see it in action as we work through our problem.

Step 1: Combine Like Terms

The first thing we want to do when solving 3q + 5 + 2q - 5 = 65 is to simplify the equation by combining like terms. Like terms are terms that have the same variable raised to the same power. In our equation, we have two terms with 'q' (3q and 2q) and two constant terms (5 and -5). Let's group them together: (3q + 2q) + (5 - 5) = 65. Now, we can combine these terms. 3q + 2q equals 5q, and 5 - 5 equals 0. So, our equation simplifies to 5q + 0 = 65, which further simplifies to 5q = 65. This step is crucial because it makes the equation much easier to work with. By combining like terms, we've reduced the complexity and brought the equation closer to a form where we can easily isolate the variable 'q'. Always remember this step when you see multiple terms with the same variable or constant terms – it's a game-changer!

Step 2: Isolate the Variable 'q'

Now that we have the simplified equation 5q = 65, our next goal is to isolate the variable 'q'. This means getting 'q' all by itself on one side of the equation. Currently, 'q' is being multiplied by 5. To undo this multiplication, we need to perform the inverse operation, which is division. We'll divide both sides of the equation by 5. This gives us (5q) / 5 = 65 / 5. On the left side, the 5s cancel out, leaving us with just 'q'. On the right side, 65 divided by 5 is 13. Therefore, we have q = 13. By performing this simple division, we've successfully isolated 'q' and found its value. Remember, the key to isolating a variable is to use the inverse operation of whatever is being done to it. This is a fundamental technique in algebra and will serve you well in solving a wide range of equations.

Step 3: Verify the Solution

We've found that q = 13, but it's always a good idea to verify our solution. This means plugging the value we found for 'q' back into the original equation to make sure it makes the equation true. Our original equation was 3q + 5 + 2q - 5 = 65. Let's substitute 13 for 'q': 3(13) + 5 + 2(13) - 5. Now, we simplify: 3 * 13 = 39, and 2 * 13 = 26. So, we have 39 + 5 + 26 - 5. Adding these numbers together, 39 + 5 is 44, and 26 - 5 is 21. Finally, 44 + 21 = 65. This matches the right side of our original equation, so our solution is correct! Verifying your solution is a crucial step because it helps you catch any mistakes you might have made along the way. It gives you confidence that you've solved the problem correctly.

The Answer

So, after simplifying and solving the equation 3q + 5 + 2q - 5 = 65, we found that the value of 'q' is 13. Therefore, the correct answer is A. q = 13.

Step-by-step Solution

To recap, here's the step-by-step solution to the equation:

  1. Combine like terms: 3q + 2q + 5 - 5 = 65 simplifies to 5q = 65.
  2. Isolate the variable: Divide both sides by 5: (5q) / 5 = 65 / 5, which gives q = 13.
  3. Verify the solution: Substitute q = 13 back into the original equation: 3(13) + 5 + 2(13) - 5 = 65, which is true.

Common Mistakes to Avoid

When solving equations like this, there are a few common mistakes that students often make. One of the biggest is forgetting to distribute properly when there are parentheses. In our equation, we didn't have parentheses, but if we did, it would be crucial to multiply the term outside the parentheses by each term inside. Another common mistake is not performing the same operation on both sides of the equation. Remember, the equation is like a balanced scale – whatever you do to one side, you must do to the other. Also, be careful with signs (positive and negative). A small mistake with a sign can throw off your entire solution. Finally, always double-check your work, especially when dealing with multiple steps. Verification, as we discussed earlier, is your best friend in catching these errors.

Practice Problems

To really master solving linear equations, you need to practice! Here are a few practice problems for you to try:

  1. 2x + 7 + 3x - 2 = 20
  2. 4y - 9 + y + 3 = 14
  3. 6z + 1 - 2z - 5 = 8

Try solving these problems using the steps we've discussed. Remember to combine like terms, isolate the variable, and verify your solution. If you get stuck, go back and review the steps we covered in this guide. The more you practice, the more confident you'll become in solving these types of equations.

Conclusion

Great job, guys! You've now learned how to simplify and solve the equation 3q + 5 + 2q - 5 = 65. We've covered the key steps: combining like terms, isolating the variable, and verifying the solution. Remember, practice is key to mastering these skills. Keep working at it, and you'll become a pro at solving linear equations. If you have any questions or want to explore more challenging problems, don't hesitate to ask. Happy solving!