Solving For 'g': A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into a common algebraic problem: solving for a variable. Specifically, we'll be tackling the equation 15 = 3(-13 + g). Don't worry if equations give you a bit of a headache; we'll break it down into easy-to-follow steps. By the end of this guide, you'll be confidently solving for 'g' and understanding the core principles behind these types of problems. Let's get started!

Understanding the Basics: Equations and Variables

Before we jump into the solution, let's quickly recap some fundamental concepts. An equation is a mathematical statement that asserts the equality of two expressions. It's like a balanced scale; whatever you do to one side, you must also do to the other to keep it balanced. In our equation, 15 = 3(-13 + g), the left side (15) must be equal to the right side (3 multiplied by the sum of -13 and g).

A variable, represented by a letter (in this case, 'g'), is a symbol that represents an unknown value. Our goal is to isolate the variable, which means getting 'g' by itself on one side of the equation. To do this, we'll use a combination of mathematical operations, ensuring that we maintain the equality throughout the process. Think of 'g' as a hidden number we're trying to uncover. The operations we use are based on the properties of equality, which state that we can add, subtract, multiply, or divide both sides of an equation by the same non-zero number without changing the solution. This is how we manipulate the equation while preserving its balance and ultimately solve for the value of our hidden number 'g'.

Mastering these basics is crucial. You'll often come across equations like this in various mathematical contexts, from basic algebra to more advanced topics. Knowing how to manipulate and solve equations is a fundamental skill that underpins your ability to understand and work with mathematical models and real-world problems. Whether you're balancing a checkbook, calculating the trajectory of a rocket, or designing a computer program, the ability to solve equations will serve you well. It's a foundational skill that opens the door to countless possibilities in mathematics and related fields, making it essential for anyone looking to deepen their understanding of numbers and their relationships. Understanding the value of variables and how they interact within equations is also a stepping stone to grasping more advanced concepts like functions, calculus, and other areas of higher mathematics.

Step-by-Step Solution: Unveiling the Value of 'g'

Alright, let's get down to business! Here's how we solve for 'g' in the equation 15 = 3(-13 + g):

Step 1: Distribute

First, we need to deal with the parentheses on the right side of the equation. We do this by distributing the 3 to both terms inside the parentheses. Remember, distributing means multiplying each term inside the parentheses by the number outside. So, we multiply 3 by -13 and 3 by 'g':

3 * -13 = -39

3 * g = 3g

This gives us a new equation: 15 = -39 + 3g

This initial step is crucial for simplifying the equation. It eliminates the parentheses and allows us to isolate the variable, 'g'. Distribution simplifies the expression, turning a single term with a set of parentheses into separate terms. This process is essentially applying the distributive property of multiplication over addition, which is a fundamental concept in algebra. By distributing, we are essentially breaking down the problem into smaller, more manageable parts, which simplifies the subsequent steps and reduces the likelihood of errors. The careful application of this rule is the first step towards a solution, transforming the original equation into a form that's easier to manipulate and solve.

Step 2: Isolate the Variable Term

Next, we need to get the term with 'g' (which is 3g) by itself on one side of the equation. To do this, we'll get rid of the -39 on the right side. We can do this by adding 39 to both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced.

15 + 39 = -39 + 39 + 3g

This simplifies to: 54 = 3g

This step is all about moving all constants to one side of the equation, leaving the variable term on the other side. By adding 39 to both sides, we use the addition property of equality, ensuring that the equality is maintained. The goal here is to transform the equation into a form where we can easily solve for 'g'. This step takes us one step closer to isolating 'g'. This is a critical move, eliminating the constant term from the side where the variable is, thus preparing us for the final step: solving for the exact value of g. The careful balance is maintained, ensuring our solution is accurate and valid.

Step 3: Solve for 'g'

Now, we have 54 = 3g. To get 'g' by itself, we need to divide both sides of the equation by 3. This isolates 'g' and reveals its value.

54 / 3 = 3g / 3

This simplifies to: 18 = g

So, the solution to the equation 15 = 3(-13 + g) is g = 18. Congratulations, you've solved for 'g'! This is the culmination of our efforts. This final step isolates 'g' by dividing both sides by the coefficient of the variable. By dividing by 3, we use the division property of equality. This isolates 'g' and gives us its precise value: 18. This is the moment we unveil the secret of the equation, the value that satisfies the original relationship. The methodical steps we've followed have led us here, showcasing the elegance of algebraic manipulation. It’s a testament to the fact that with careful application of mathematical principles, complex-looking equations become remarkably solvable.

Verification: Ensuring the Correct Solution

It's always a good practice to check your answer. Let's substitute g = 18 back into the original equation 15 = 3(-13 + g):

15 = 3(-13 + 18)

15 = 3(5)

15 = 15

The equation holds true! This confirms that our solution, g = 18, is correct. Verification is essential. It provides assurance, by confirming the answer adheres to the original statement. It catches any potential errors made during the solving process. Substituting the value back into the original equation ensures that the equality remains valid. It’s like double-checking your work, ensuring both sides balance. This process assures accuracy and confirms our understanding of the underlying principles. This step is about reinforcing confidence and ensuring that the calculated solution is correct. It underscores the precision needed in mathematics and how each step builds on the previous one to reach a valid answer.

Tips and Tricks for Solving Equations

Here are some helpful tips to keep in mind when solving equations:

• Stay Organized: Write out each step clearly and neatly. This helps you avoid mistakes and makes it easier to track your progress.
• Double-Check Your Work: After solving, always substitute your answer back into the original equation to verify that it's correct.
• Practice Regularly: The more you practice, the more comfortable and confident you'll become with solving equations. Work through a variety of examples to build your skills.
• Understand the Properties of Equality: Make sure you understand the addition, subtraction, multiplication, and division properties of equality. These are the foundation of solving equations.
• Simplify First: Before you start isolating the variable, simplify each side of the equation as much as possible. This can often make the problem easier to solve.

These tips are valuable. They are designed to improve efficiency and reduce errors. Staying organized promotes clarity and makes it easier to spot errors. Regular practice cements the concepts and allows you to build confidence. Understanding equality properties is fundamental for equation solving. Simplifying equations early makes solving easier. These are essential for mastering algebraic manipulation.

Conclusion: Your Journey into Algebra

And there you have it! You've successfully solved for 'g' in the equation. You've learned the fundamental steps, including distribution, isolating the variable term, and solving for the variable. You've also verified your answer and learned some helpful tips. Remember, practice is key to mastering algebra. As you continue to work through more equations, you'll find that these steps become second nature. Keep exploring the world of mathematics and don't be afraid to tackle new challenges. Each problem solved is a step forward in your mathematical journey. With persistence, you'll develop the skills to solve even more complex equations and expand your mathematical horizons.

Keep practicing, keep learning, and enjoy the process! You’ve taken a solid step toward improving your math skills. Keep seeking new knowledge and don't hesitate to seek help when needed. The world of mathematics is vast and rewarding, offering endless opportunities for growth and discovery. Embrace the challenge and watch your mathematical abilities flourish.

For further study and practice, check out these resources:

• Khan Academy (https://www.khanacademy.org/): Khan Academy offers free, comprehensive lessons and practice exercises on algebra and other math topics. They provide a structured learning path with video tutorials, practice problems, and progress tracking.