3g–10=4+8g–7g

When dealing with algebraic equations, one of the primary goals is to isolate the variable on one side of the equation. This allows for easier solutions and a better understanding of the problem. In this article, we will break down the equation 3g–10=4+8g–7g and provide a detailed, step-by-step explanation of how to solve it. Whether you are a student learning algebra or just someone looking to brush up on their math skills, this article will guide you through the process in a clear, easy-to-follow manner.

Introduction to Algebraic Equations

Algebraic equations are mathematical statements that assert the equality of two expressions. These equations often involve variables, constants, and operations such as addition, subtraction, multiplication, and division. The main goal in solving an equation is to find the value of the variable that makes the equation true.

In this case, the equation is: 3g−10=4+8g−7g3g – 10 = 4 + 8g – 7g

Our job is to solve for g, which is the unknown variable.

Understanding the Terms in the Equation

Before diving into the solution process, let’s break down the equation into its individual terms:

  • 3g: This is the first term on the left side of the equation. It represents a value that is three times the unknown variable g.
  • -10: This is a constant term on the left side of the equation. It is subtracted from the 3g term.
  • 4: This is a constant term on the right side of the equation.
  • 8g and -7g: These are terms that involve the variable g on the right side of the equation. 8g represents eight times g, and -7g represents seven times g with a negative sign.

By simplifying the equation step by step, we will solve for the value of g.

Step 1: Simplify the Right-Hand Side

The first step in solving the equation is to simplify the right-hand side of the equation. On the right side, we have 8g – 7g. Since both terms involve the variable g, we can combine them.

8g−7g=g8g – 7g = g

Now, the equation looks like this:

3g−10=4+g3g – 10 = 4 + g

Step 2: Move All Terms Involving g to One Side

Next, we want to isolate g on one side of the equation. To do this, we need to move all terms involving g to one side of the equation and the constants to the other side. We start by subtracting g from both sides of the equation to eliminate g on the right side.

3g−g−10=43g – g – 10 = 4

Simplifying this gives:

2g−10=42g – 10 = 4

Step 3: Move the Constant Term to the Other Side

Now that we have 2g on the left side, we need to move the constant term -10 to the other side. To do this, we add 10 to both sides of the equation.

2g=4+102g = 4 + 10

Simplifying this gives:

2g=142g = 14

Step 4: Solve for g

At this point, we have 2g = 14. To solve for g, we simply divide both sides of the equation by 2.

g=142g = \frac{14}{2}

Simplifying this gives:

g=7g = 7

So, the solution to the equation 3g – 10 = 4 + 8g – 7g is g = 7.

Verifying the Solution

It’s always a good idea to verify the solution by substituting the value of g back into the original equation. Let’s substitute g = 7 into the original equation:

3(7)−10=4+8(7)−7(7)3(7) – 10 = 4 + 8(7) – 7(7)

Simplifying both sides:

21−10=4+56−4921 – 10 = 4 + 56 – 49

On the left side:

21−10=1121 – 10 = 11

On the right side:

4+56−49=114 + 56 – 49 = 11

Both sides are equal, so the solution g = 7 is correct.

Tips for Solving Similar Equations

Solving equations like 3g – 10 = 4 + 8g – 7g is a skill that improves with practice. Here are some tips for solving similar equations:

  1. Simplify both sides of the equation: Always begin by simplifying the terms on both sides. Combine like terms to make the equation easier to work with.
  2. Isolate the variable: Move all terms involving the variable to one side and constants to the other side. This makes it easier to solve for the variable.
  3. Check your work: After solving for the variable, substitute the value back into the original equation to ensure that both sides are equal.
  4. Practice: The more you practice solving algebraic equations, the more confident you will become.

Common Mistakes to Avoid

While solving algebraic equations, it’s easy to make some common mistakes. Here are a few to watch out for:

  1. Incorrectly combining like terms: Always ensure that you are combining like terms correctly. For example, in the equation 8g – 7g, make sure to subtract the coefficients (8 – 7 = 1).
  2. Forgetting to apply operations to both sides of the equation: When you add, subtract, multiply, or divide on one side of the equation, remember to perform the same operation on the other side.
  3. Misinterpreting negative signs: Negative signs can sometimes be tricky. Always check that you are applying them correctly, especially when dealing with subtraction.

Why It’s Important to Master Solving Equations

Understanding how to solve equations is fundamental to mastering algebra and higher-level mathematics. Algebraic equations are the foundation for many real-world applications, including physics, engineering, economics, and computer science. By learning how to solve equations efficiently, you will develop problem-solving skills that are valuable in many fields.

Applications of Algebraic Equations

Algebraic equations are not just abstract concepts; they have real-world applications in many areas of life. Here are a few examples:

  • Finance: Algebraic equations can be used to calculate interest, investments, and loans. For example, you might use equations to determine how much interest you will earn on a savings account.
  • Physics: In physics, equations are used to model various phenomena, such as motion, forces, and energy. Solving these equations is crucial for predicting and understanding how objects move and interact.
  • Engineering: Engineers use algebraic equations to design structures, electrical circuits, and machines. Understanding how to solve these equations is essential for creating safe and functional designs.

Conclusion

Solving algebraic equations, such as 3g – 10 = 4 + 8g – 7g, is a valuable skill that will serve you in many areas of mathematics and beyond. By breaking the problem down step by step and isolating the variable, we were able to determine that g = 7 is the solution.

As you practice more equations, you’ll gain confidence and improve your problem-solving abilities. Remember to simplify the terms, move the variables to one side, and check your work to avoid common mistakes. Algebra is an essential tool that helps you understand and solve a wide variety of problems in the world around you.

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