– 3g–10= – 6g+8

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11-03-2025

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Solving the Equation: -3g - 10 = -6g + 8 – A Step-by-Step Guide

This article explores the solution to the algebraic equation -3g - 10 = -6g + 8, providing a detailed step-by-step process and offering insights into the underlying mathematical principles. We'll delve into the techniques used to solve such equations, and explore practical applications where similar equations might arise. While this specific equation doesn't directly come from a ScienceDirect article (as it's a fundamental algebra problem), the principles used to solve it are foundational to many scientific and engineering calculations discussed within such publications.

Understanding the Equation

The equation -3g - 10 = -6g + 8 is a linear equation in one variable (g). The goal is to find the value of 'g' that makes the equation true. This involves isolating 'g' on one side of the equation through a series of algebraic manipulations. These manipulations are based on the properties of equality: whatever operation you perform on one side of the equation, you must perform on the other to maintain balance.

Solving the Equation: Step-by-Step

1. Add 6g to both sides: Our first step is to gather all the terms containing 'g' on one side of the equation. We can achieve this by adding 6g to both sides:

   -3g - 10 + 6g = -6g + 8 + 6g

   This simplifies to:

   3g - 10 = 8

2. Add 10 to both sides: Next, we want to isolate the term with 'g' by removing the constant term (-10). We add 10 to both sides:

   3g - 10 + 10 = 8 + 10

   This simplifies to:

   3g = 18

3. Divide both sides by 3: Finally, to solve for 'g', we divide both sides of the equation by 3:

   3g / 3 = 18 / 3

   This gives us the solution:

   g = 6

Verification:

To verify our solution, we substitute g = 6 back into the original equation:

-3(6) - 10 = -6(6) + 8

-18 - 10 = -36 + 8

-28 = -28

Since both sides are equal, our solution g = 6 is correct.

Practical Applications and Extensions

Linear equations like this are fundamental to many areas:

• Physics: Calculating velocities, accelerations, and forces often involves solving linear equations. For example, determining the time it takes for an object to fall a certain distance under gravity involves a linear equation considering acceleration due to gravity. (See Halliday, Resnick, and Walker, "Fundamentals of Physics," which is frequently cited in ScienceDirect publications and uses similar equation-solving techniques).

• Engineering: Designing structures, analyzing circuits, and modeling systems often require solving systems of linear equations (which build upon the principles demonstrated here). For instance, calculating the stresses in a bridge member might involve setting up and solving several equations that include variables for forces and angles. (This relates to numerous articles in ScienceDirect concerning structural mechanics and engineering).

• Economics: Linear equations are used to model supply and demand curves, helping economists to predict market behavior. The equilibrium point where supply equals demand is found by solving a linear equation. (See numerous economic modelling articles in ScienceDirect that rely on these fundamental principles).

• Computer Science: Linear equations form the basis for many algorithms in computer graphics, machine learning, and optimization problems. Solving these equations efficiently is crucial for the performance of these algorithms. (Look at algorithms for solving systems of linear equations in ScienceDirect publications relating to numerical methods and computational science).

Expanding the Concept: More Complex Equations

While this example is a relatively simple linear equation, the principles involved can be extended to more complex scenarios:

• Equations with fractions: Equations involving fractions can be solved by finding a common denominator and simplifying.

• Equations with parentheses: Equations containing parentheses require distributing the term outside the parenthesis before combining like terms and solving for the variable.

• Systems of linear equations: Multiple equations with the same variables are solved using techniques like substitution or elimination.

Conclusion:

Solving the equation -3g - 10 = -6g + 8 demonstrates a fundamental process in algebra. The step-by-step approach, using the properties of equality, provides a clear and concise method to isolate the variable and find the solution. This seemingly simple equation underpins much more complex mathematical models and has wide-ranging applications across numerous scientific and engineering disciplines. Understanding this basic procedure is crucial for tackling more advanced mathematical challenges encountered in various fields. While not directly sourced from a specific ScienceDirect article, the methodology and underlying principles are consistently applied and expanded upon within countless publications available on that platform. This article serves to highlight the importance of mastering these foundational algebraic skills.