Solve Equations: Complete The Process & Simplify

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Solve Equations: Complete the Process & Simplify

Alright, guys, let's dive into some equation solving fun! We're gonna break down the process step-by-step, fill in those pesky missing terms, and simplify fractions like pros. This is all about making sure we understand how to manipulate equations and find the value of our variable, "a" in this case. So, grab your pencils, and let's get started. We will learn how to approach the equation, how to isolate the variable, and how to perform mathematical operations to arrive at the solution. Let's start with the equation and fill in the missing terms in the process of solving it.

Step-by-Step Equation Solving

Our goal here is to carefully work through the given equation, completing each step and making sure we understand what's happening. The original problem is: -6(2a + 7) = -18a - 6. We'll use this to build a solid foundation, which makes it easy to handle more complex equations. Every single step is important, and we'll break them down. It's like a recipe; we need all the ingredients and follow the instructions to get the final result. Remember to take it slow. If we get the small steps correct, the big picture will eventually reveal itself.

First, we distribute the -6 across the terms inside the parentheses. The first term is -6 * 2a which equals -12a. Next, we have -6 * 7, which is -42. So far, our equation looks like this: -12a - 42 = -18a - 6. Now, let's compare this with the next line provided in the problem: -12a - \_ = -18a - 6. Based on the first step we made, it is easy to see that the missing value is 42. So, we'll write 42 in the blank. The equation becomes: -12a - 42 = -18a - 6. After that, we need to isolate the variable "a". In order to do that, we move all the "a" terms to one side of the equation and the constant terms to the other side. So, we add 18a to both sides to get -12a + 18a - 42 = -18a + 18a - 6. Simplifying the equation, we get 6a - 42 = -6. The next line given to us in the problem is: \_ - 42 = -6. Therefore, the blank should be 6a. The equation becomes: 6a - 42 = -6. We are getting closer to the solution. Now, we add 42 to both sides of the equation. We add 42 to -6 to get 36. Therefore, the next line should be 6a = 36. Finally, we divide both sides by 6 to isolate "a". So, a = 6. We successfully solved the equation and filled in all the blanks. We can apply this method to other mathematical problems. The most important thing here is to understand the concepts behind these problems and practice doing them.

Filling in the Missing Terms and Simplifying

Let's go back and work through this problem again, but this time, we will fill in the missing terms and simplify fractions. Our original equation is -6(2a + 7) = -18a - 6. The first thing we need to do is distribute that -6 across the terms within the parenthesis. This means we multiply -6 by both 2a and 7. So, -6 * 2a = -12a and -6 * 7 = -42. This gives us -12a - 42 = -18a - 6. That is the first step! In the next step, we have -12a - \_ = -18a - 6. So, the missing term is 42. Our equation becomes -12a - 42 = -18a - 6. Now, we move onto the next step. To get to the next step, we add 18a to both sides, which makes the equation 6a - 42 = -6. So, in the problem, \_ - 42 = -6, the missing term is 6a. After that, we add 42 to both sides of the equation, to get 6a = 36. And finally, we divide both sides by 6, which gives us a = 6. And that is our final result. Every step has led us closer to the correct answer. The process is easy, just follow the right steps.

The Complete Solution

Here is the complete process with all the missing terms filled in. We are gonna put everything together, making sure we haven't missed anything. This is what the solved equation looks like:

  • -6(2a + 7) = -18a - 6 (Starting Equation)
  • -12a - 42 = -18a - 6 (Distribute -6)
  • 6a - 42 = -6 (Add 18a to both sides)
  • 6a = 36 (Add 42 to both sides)
  • a = 6 (Divide both sides by 6)

As you can see, we started with the original equation and worked through it step-by-step. Remember, the key is to be meticulous. Check every single calculation, and ensure that you're correctly applying the rules of algebra. It's often helpful to double-check your work, particularly when dealing with negative signs and fractions. In mathematics, small mistakes can lead to big problems, so practice good habits! If we take our time and carefully work through each stage, we can avoid any errors. And if you do make a mistake, don't worry about it! It's all part of the learning process. The key is to learn from it and try again. Practice makes perfect. Don't be afraid to redo the steps. After several rounds, you can solve the problem easily.

Explanation of Each Step

Let's break down each step and explain why we did what we did. In the first step, we distributed the -6. This is based on the distributive property, which tells us that we must multiply the term outside the parenthesis by each term inside the parenthesis. Then, in the second step, we added 18a to both sides. By adding 18a to both sides, we want to isolate the variable on one side. Remember that in mathematics, whatever we do to one side of an equation, we must do to the other side to keep it balanced. This rule applies throughout the whole process. The same rule applies when we added 42 to both sides. The final step is to divide both sides by 6, in order to isolate "a". When we have "a" on its own on one side of the equation, we know that the value is the solution of the equation. It's all about keeping things balanced and following the established mathematical rules.

Simplifying Fractions (If Applicable)

In our particular problem, we didn't have to deal with simplifying fractions. But in some equations, you'll definitely run into fractions, and it's essential to know how to handle them. For example, if we had ended up with something like a = 12/2, then we would need to simplify that fraction. So, 12/2 would simplify to 6. If you have fractions, always look to simplify them. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. Dividing both the numerator and denominator by their greatest common factor (GCF) reduces a fraction to its simplest form. When simplifying, always try to express fractions in their lowest terms.

Conclusion

So, there you have it, guys. We've gone through the process of solving the equation together, filled in the missing terms, and discussed simplifying fractions. Remember, solving equations is a fundamental skill in math, and with practice, you'll become more and more comfortable with it. Each equation has a unique solution. Always practice doing different equations. Keep practicing, and don't hesitate to ask for help if you need it. You got this!