Solve For G: Step-by-Step Guide With Examples

Hey everyone! Today, we're diving into a classic algebra problem: solving for the variable g. Don't worry, even if equations make you sweat a little, we'll break it down step by step so you can conquer it with confidence. We'll focus on the equation -4g - (-2g) = 300. This type of problem pops up all the time in math, science, and even everyday life, so mastering it is a fantastic skill to have. Let's get started and turn that equation into a piece of cake!

Understanding the Equation: The Foundation of Success

Before we jump into the nitty-gritty of solving for g, let's take a moment to really understand what this equation is telling us. The equation, -4g - (-2g) = 300, is a linear equation. This means it involves a variable (g in our case) raised to the power of 1. Linear equations are the building blocks of algebra, and they represent a straight-line relationship when graphed. Think of it like a balancing scale: the left side of the equation must always equal the right side. Our goal is to isolate g on one side of the equation, so we can see exactly what value makes the equation true. To do that, we need to understand the different parts of the equation and how they interact. We have coefficients (the numbers multiplying the variable), constants (the numbers on their own), and the variable itself. The operations (subtraction in this case) also play a crucial role. By understanding these elements, we can strategically manipulate the equation to get g all by itself. Remember, each step we take is like carefully adjusting the weights on our balancing scale to maintain equilibrium. We will simplify, combine like terms, and use inverse operations to solve g. So, let's get started with the first step: simplifying the equation.

Step 1: Simplifying the Equation – Taming the Negatives

The first thing we need to tackle in our equation, -4g - (-2g) = 300, is that double negative. Double negatives can be a bit confusing, but remember this golden rule: subtracting a negative number is the same as adding a positive number. So, “- (-2g)” becomes “+ 2g”. This simplifies our equation to -4g + 2g = 300. See? Already it looks a little less intimidating! This step is crucial because it sets us up for combining like terms, which is the next step in isolating g. Think of it like decluttering your workspace before starting a project – it makes everything easier to manage. By simplifying the equation, we've made it more manageable and easier to work with. This is a common strategy in algebra: break down complex problems into smaller, more digestible steps. Now that we've tamed the negatives, we're ready to move on to combining those “g” terms. This is where we'll start to see the value of g come into focus. So, let's jump into the next step and continue our journey to solving for g. Remember, each step brings us closer to the final answer!

Step 2: Combining Like Terms – Gathering the 'g's

Now that we've simplified the equation to -4g + 2g = 300, it's time to combine those “g” terms. Think of it like grouping similar objects together. We have -4 lots of g and we're adding 2 lots of g. What does that give us? It's like saying, “I owe you 4 apples, but then I give you 2 back. How many do I still owe?” The answer is 2 apples, or in our case, -2g. So, -4g + 2g simplifies to -2g. This means our equation is now -2g = 300. We're getting closer! We've successfully combined the “g” terms, making the equation even simpler. This step is essential because it reduces the number of terms we need to deal with, bringing us closer to isolating g. By combining like terms, we've streamlined the equation, making it easier to solve. It's like condensing a recipe – we've kept the important ingredients but removed the unnecessary fluff. Now, we're just one step away from finding the value of g. The next step involves using the inverse operation to isolate g completely. So, let's move on to the final step and claim our victory!

Step 3: Isolating 'g' – The Grand Finale

We've arrived at the final step! Our equation is now -2g = 300. Remember, our goal is to get g all by itself on one side of the equation. Right now, g is being multiplied by -2. To undo this multiplication, we need to use the inverse operation: division. We'll divide both sides of the equation by -2. This is crucial because what we do to one side of the equation, we must do to the other to maintain the balance. So, we divide both -2g and 300 by -2. On the left side, -2g divided by -2 is simply g (the -2s cancel out). On the right side, 300 divided by -2 is -150. Therefore, our solution is g = -150. We did it! We've successfully isolated g and found its value. This step is the culmination of all our hard work. By using the inverse operation, we've untangled g from its coefficient and revealed its true value. It's like solving a puzzle – the final piece clicks into place, and the picture is complete. Now that we've found our solution, it's always a good idea to check our work to make sure we haven't made any mistakes. So, let's move on to the final section: verifying our solution.

Verifying the Solution: The Seal of Approval

We've solved for g, and we think we have the answer: g = -150. But before we celebrate too much, let's make sure our solution is correct. This is where verification comes in. It's like double-checking your work on an exam – it's always a good idea to catch any errors. To verify our solution, we'll substitute -150 back into the original equation: -4g - (-2g) = 300. Replacing g with -150, we get: -4*(-150) - (-2*(-150)) = 300. Now, let's simplify. -4 multiplied by -150 is 600. -2 multiplied by -150 is 300, and then we have the subtraction, so it's -300. So, the left side of the equation becomes 600 - 300, which equals 300. And that's exactly what the right side of the equation is! Since both sides of the equation are equal when we substitute g = -150, our solution is correct. Excellent! Verification is a vital step in problem-solving. It ensures that our hard work has paid off and that we can confidently say we've found the right answer. It's like having a seal of approval on our solution. By verifying our solution, we've gained confidence in our answer and in our problem-solving skills. We've not only solved for g but also proven that our solution is accurate. So, congratulations! You've successfully navigated this algebra problem from start to finish. Now, let's recap what we've learned.

Conclusion: Mastering the Art of Solving for 'g'

So, guys, we've successfully solved for g in the equation -4g - (-2g) = 300, and the answer is g = -150. We started by understanding the equation, then we simplified it by tackling the double negative. Next, we combined like terms to make the equation even cleaner. We then isolated g by using the inverse operation of division. And finally, we verified our solution to make sure we got it right. This process highlights the key steps in solving linear equations: simplify, combine like terms, isolate the variable, and verify. These steps are like a roadmap for tackling algebra problems. By mastering them, you'll be well-equipped to solve a wide range of equations. Remember, practice makes perfect. The more you work with equations, the more comfortable you'll become with the process. So, don't be afraid to tackle new problems and challenge yourself. You've got the tools and the knowledge to succeed! Solving for variables like g is a fundamental skill in mathematics and has applications in many other fields. From science and engineering to finance and economics, the ability to manipulate equations is invaluable. So, keep practicing, keep learning, and keep exploring the world of mathematics. You never know what amazing discoveries you'll make along the way!