Solving -2a - 7 = -13 A Step By Step Guide
Hey guys! Today, we're diving into the world of algebra and tackling a linear equation. Don't worry if equations seem intimidating at first; we're going to break it down step-by-step, making it super easy to understand. Our mission today is to solve the equation -2a - 7 = -13. We'll walk through each step, explaining the logic behind it, so you not only get the answer but also grasp the methods involved. Solving equations is a fundamental skill in mathematics, and mastering it opens doors to more advanced topics. So, let's get started and turn this equation into a piece of cake!
Understanding Linear Equations
Before we jump into solving, let's quickly touch on what a linear equation actually is. A linear equation is basically an algebraic equation where the highest power of the variable (in our case, 'a') is 1. Think of it as a straight line when you graph it – hence the name 'linear'. These equations involve constants (numbers) and variables (letters representing unknown values) connected by mathematical operations like addition, subtraction, multiplication, and division. Our goal when solving a linear equation is to isolate the variable on one side of the equation, revealing its value. This is done by carefully applying inverse operations to both sides of the equation, ensuring we maintain balance and equality. Now that we've got a basic understanding, let's dive into the specific steps for solving our equation, -2a - 7 = -13. Remember, each step is like a puzzle piece, bringing us closer to the final solution. Understanding the 'why' behind each step is just as important as the 'how', so let's get started!
Step 1: Isolating the Term with the Variable
The first step in solving -2a - 7 = -13 is to isolate the term that contains our variable, which is -2a. Currently, we have “-7” hanging out on the same side, and we need to get rid of it. The key here is to use the inverse operation. Since we are subtracting 7, the inverse operation is addition. We're going to add 7 to both sides of the equation. Why both sides? Because in an equation, think of the equals sign (=) as a balance scale. Whatever you do to one side, you must do to the other to keep it balanced. If we only added 7 to the left side, the equation would no longer be true. So, let's add 7 to both sides:
-2a - 7 + 7 = -13 + 7
On the left side, -7 and +7 cancel each other out, leaving us with just -2a. On the right side, -13 + 7 equals -6. Our equation now looks like this:
-2a = -6
We've successfully isolated the term with our variable! This is a significant step forward. We're one step closer to finding the value of 'a'. Remember, the goal is always to get 'a' all by itself on one side of the equation. Next, we'll tackle the coefficient that's attached to 'a'.
Step 2: Solving for the Variable
Okay, we've reached the final step in solving for 'a' in the equation -2a = -6. We've successfully isolated the term containing 'a', and now we just need to get 'a' by itself. Currently, 'a' is being multiplied by -2. To undo this multiplication, we'll use the inverse operation, which is division. We're going to divide both sides of the equation by -2. Again, it's crucial to remember the balance principle: what we do to one side, we must do to the other.
So, let's divide both sides by -2:
-2a / -2 = -6 / -2
On the left side, -2 divided by -2 equals 1, so we're left with just 'a'. On the right side, -6 divided by -2 equals 3 (remember, a negative divided by a negative is a positive). Our equation now looks like this:
a = 3
We've done it! We've successfully solved for 'a'. The value of 'a' that makes the equation true is 3. This is the solution to our equation. We've isolated the variable and found its value by carefully applying inverse operations. But, as any good mathematician knows, it's always a good idea to check our work!
Step 3: Checking the Solution
Alright, we've found our solution: a = 3. But before we declare victory, it's super important to check our work. This ensures we haven't made any sneaky errors along the way. To check our solution, we're going to substitute the value we found for 'a' (which is 3) back into the original equation, -2a - 7 = -13. If the equation holds true after the substitution, we know we've got the right answer. If not, we need to go back and carefully review our steps to find any mistakes.
Let's substitute a = 3 into the original equation:
-2(3) - 7 = -13
Now, let's simplify the left side of the equation. First, we multiply -2 by 3, which gives us -6:
-6 - 7 = -13
Next, we subtract 7 from -6, which gives us -13:
-13 = -13
Boom! The left side of the equation equals the right side of the equation. This confirms that our solution, a = 3, is indeed correct. We've successfully solved the equation and verified our answer. Checking your work is a crucial habit in mathematics, and it can save you from making simple errors. So, always take that extra step to ensure your solution is accurate. Now, let's recap what we've learned.
Conclusion: Mastering Linear Equations
So, guys, we've successfully solved the equation -2a - 7 = -13, and we found that a = 3. We didn't just get the answer, though; we also walked through the process, understanding each step along the way. We started by understanding what a linear equation is, then we systematically isolated the variable using inverse operations. Remember, the key is to maintain balance by performing the same operation on both sides of the equation. Finally, we emphasized the importance of checking our solution to ensure accuracy.
Solving linear equations is a fundamental skill in algebra, and it's something you'll use again and again in more advanced math courses. By mastering these steps, you're building a strong foundation for future success. Don't be afraid to practice and tackle more equations! The more you practice, the more comfortable and confident you'll become. Keep up the great work, and remember, math can be fun and rewarding when you break it down into manageable steps. You've got this!
