Solving Equations Step-by-Step: A Complete Guide

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Solving Equations Step-by-Step: A Complete Guide

Hey guys! Today, we're going to break down the process of solving the equation 2(-4q - 15) - 20 = -2 step by step. We'll fill in the missing terms, simplify fractions, and make sure everything is crystal clear. So, let's dive right in and get those algebraic muscles working! This is a common type of problem in mathematics, particularly in algebra, and mastering it can significantly boost your problem-solving skills. Understanding how to manipulate equations, apply the distributive property, and combine like terms are fundamental concepts that extend to more complex mathematical problems. Whether you're a student tackling homework or someone brushing up on their math skills, this guide will provide a comprehensive walkthrough.

Breaking Down the Equation: 2(-4q - 15) - 20 = -2

Our initial equation is 2(-4q - 15) - 20 = -2. The first thing we need to do is tackle the parentheses. Remember the distributive property? It's like giving everyone in the parentheses a little gift from the number outside. In this case, the '2' needs to be multiplied by both '-4q' and '-15'.

  • Applying the Distributive Property: This step is crucial as it simplifies the equation and allows us to combine like terms later. The distributive property states that a(b + c) = ab + ac. In our case, 'a' is 2, 'b' is -4q, and 'c' is -15. Multiplying 2 by -4q gives us -8q, and multiplying 2 by -15 gives us -30. So, 2(-4q - 15) becomes -8q - 30.
  • Common Mistakes to Avoid: One common mistake is forgetting to distribute to all terms inside the parentheses. Always double-check that you've multiplied the outside number by every term inside. Another mistake is mishandling the signs. Remember that a positive number times a negative number is negative, and a negative number times a negative number is positive. Precision with signs is key to getting the correct answer.
  • Why This Step Matters: Mastering the distributive property is essential for solving more complex equations. It's a foundational concept that appears in various algebraic contexts, such as factoring, expanding polynomials, and simplifying expressions. By understanding and applying this property correctly, you'll be well-equipped to tackle more challenging problems.

Step 1: Apply the Distributive Property

Okay, so let's apply the distributive property to 2(-4q - 15). We multiply 2 by -4q and then 2 by -15. This gives us:

2 * -4q = -8q
2 * -15 = -30

So, the first missing term is -8q. Our equation now looks like this:

-8q - 30 - 20 = -2
  • The Importance of Showing Your Work: It's always a good practice to show each step in your solution. This not only helps you avoid errors but also makes it easier to track your progress. Plus, if you do make a mistake, you can quickly identify where you went wrong. Writing out each step also solidifies your understanding of the process, making it easier to recall later.
  • Tips for Staying Organized: When solving equations, keep your work neat and organized. Write each step clearly and align the equals signs. This makes it easier to read and check your work. Use arrows or other visual cues to show the steps you're taking. Organization is a key skill in mathematics, and it's something that improves with practice.
  • Checking Your Distribution: After applying the distributive property, take a moment to double-check your work. Make sure you've multiplied correctly and that you've accounted for any negative signs. A quick check here can save you from making errors later in the problem. Think of it as a mini-review point in your solution process.

Step 2: Combine the Terms

Next up, we need to combine the constant terms on the left side of the equation. We have -30 and -20. Adding these together gives us:

-30 + (-20) = -50

So, the equation becomes:

-8q - 50 = -2
  • Understanding Like Terms: Like terms are terms that have the same variable raised to the same power (or are constants). In this equation, -30 and -20 are like terms because they are both constants. Combining like terms simplifies the equation, making it easier to solve. Recognizing and combining like terms is a fundamental skill in algebra.
  • Order of Operations: Remember the order of operations (PEMDAS/BODMAS): Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. In this step, we're focusing on addition and subtraction, specifically combining constants. Following the order of operations helps ensure that you solve the equation correctly.
  • Why Combining Terms is Important: Combining like terms reduces the complexity of the equation. It's like decluttering your workspace before starting a project. By simplifying the equation, you reduce the chances of making errors and make the next steps clearer. This step is a crucial part of efficient problem-solving.

Step 3: Isolate the Variable

Now, we want to get the term with 'q' by itself on one side of the equation. To do this, we need to get rid of the -50. We can do this by adding 50 to both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep the equation balanced!

-8q - 50 + 50 = -2 + 50

This simplifies to:

-8q = 48
  • The Golden Rule of Equations: The principle of adding the same value to both sides is a cornerstone of solving equations. It's like the golden rule: treat both sides equally. This maintains the balance of the equation and ensures that you're working towards the correct solution. Keep this rule in mind as you solve any equation.
  • Why Isolation is Key: Isolating the variable is the primary goal in solving equations. It's like peeling back the layers of an onion to get to the core. Once you've isolated the variable, you're just one step away from finding its value. This step is where the equation starts to reveal its solution.
  • Checking Your Work: After adding 50 to both sides, take a moment to check your work. Make sure you've added it correctly and that the equation is still balanced. A quick check here can prevent errors in the next step. Accuracy is crucial, so always double-check your calculations.

Step 4: Solve for q

Finally, to solve for 'q', we need to get rid of the -8 that's multiplying it. We do this by dividing both sides of the equation by -8:

-8q / -8 = 48 / -8

This gives us:

q = -6

So, the final answer is q = -6. Congrats, you've solved the equation!

  • The Division Principle: Just like adding or subtracting the same value from both sides, dividing both sides by the same non-zero value keeps the equation balanced. This is another fundamental principle in solving equations. Understanding and applying this principle correctly is essential for finding the solution.
  • Checking Your Solution: Always, always, always check your solution! Plug the value you found for 'q' back into the original equation to make sure it works. If the equation holds true, then you know you've solved it correctly. This is the best way to ensure accuracy and build confidence in your problem-solving skills.
  • Practice Makes Perfect: Solving equations is like riding a bike—it gets easier with practice. The more you practice, the more comfortable you'll become with the steps and the faster you'll be able to solve problems. So, keep practicing, and don't get discouraged if you make mistakes. Mistakes are learning opportunities!

Wrapping Up

So, to recap, we've walked through the complete process of solving the equation 2(-4q - 15) - 20 = -2. We applied the distributive property, combined like terms, isolated the variable, and solved for 'q'. Remember, guys, the key is to take it one step at a time, show your work, and always check your answer. You've got this!

  • Key Takeaways: Review the main steps: apply the distributive property, combine like terms, isolate the variable, and solve for the variable. Understanding these steps is crucial for solving various algebraic equations. Keep these steps in mind as you tackle future problems.
  • Further Practice: Try solving similar equations on your own. Look for practice problems in your textbook or online. The more you practice, the more confident you'll become in your equation-solving abilities. Challenge yourself with increasingly complex problems to further develop your skills.
  • The Bigger Picture: Remember, solving equations is not just about getting the right answer. It's about developing critical thinking and problem-solving skills that are valuable in many areas of life. Math is like a muscle; the more you use it, the stronger it gets. Keep exercising your math muscles, and you'll be amazed at what you can achieve! Now you are able to solve equations step-by-step like a pro!.

Keep practicing, and you'll be solving equations like a pro in no time!