10d= – 10d–8( – 2d–9)

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11-03-2025

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Solving the Equation: 10d = -10d - 8(-2d - 9) – A Step-by-Step Guide

This article provides a comprehensive walkthrough of solving the algebraic equation 10d = -10d - 8(-2d - 9). We'll break down the process step-by-step, explain the underlying principles, and offer practical examples to solidify your understanding. While this specific equation doesn't directly appear in a readily available ScienceDirect article, the methods used to solve it are fundamental concepts covered extensively in their mathematical resources. We'll leverage these fundamental principles to analyze and solve the problem.

Understanding the Equation

The equation 10d = -10d - 8(-2d - 9) is a linear equation in one variable (d). Our goal is to isolate 'd' on one side of the equation to find its value. This involves applying the rules of algebra, including the order of operations (PEMDAS/BODMAS) and the properties of equality.

Step-by-Step Solution

1. Distribute the -8: The first step involves distributing the -8 to both terms inside the parentheses:

   -8(-2d - 9) = (-8)(-2d) + (-8)(-9) = 16d + 72

   Our equation now becomes:

   10d = -10d + 16d + 72

2. Combine Like Terms: Combine the terms with 'd' on the right-hand side of the equation:

   -10d + 16d = 6d

   The equation simplifies to:

   10d = 6d + 72

3. Isolate the Variable: To isolate 'd', we need to get all terms containing 'd' on one side of the equation and all constant terms on the other. Subtract 6d from both sides:

   10d - 6d = 6d + 72 - 6d

   This simplifies to:

   4d = 72

4. Solve for d: Finally, divide both sides by 4 to solve for 'd':

   4d / 4 = 72 / 4

   Therefore:

   d = 18

Verification

To verify our solution, substitute d = 18 back into the original equation:

10(18) = -10(18) - 8(-2(18) - 9)

180 = -180 - 8(-36 - 9)

180 = -180 - 8(-45)

180 = -180 + 360

180 = 180

The equation holds true, confirming that our solution, d = 18, is correct.

Expanding on the Concepts: Connecting to ScienceDirect Principles

The techniques used to solve this equation are fundamental to various mathematical concepts found in countless ScienceDirect articles related to mathematics, physics, engineering, and other quantitative fields. These include:

• Linear Algebra: Linear equations form the basis of linear algebra, a crucial field in many scientific disciplines. ScienceDirect hosts numerous papers on linear algebra, covering topics such as matrix operations, vector spaces, and systems of linear equations. Understanding how to solve single linear equations like the one above is a prerequisite for tackling more complex systems.

• Mathematical Modeling: Linear equations are frequently used to create mathematical models of real-world phenomena. For example, in physics, the equation might represent a simple model of motion or force. Solving such equations is essential to making predictions and drawing conclusions from the model. ScienceDirect articles in various fields use this kind of modelling extensively.

• Numerical Methods: While this specific equation can be solved analytically, more complex equations may require numerical methods for solutions. ScienceDirect features a vast body of literature on numerical methods, including techniques for approximating solutions to equations that cannot be solved exactly.

Practical Applications and Extensions

This seemingly simple equation can be extended to model various real-world scenarios. For example:

• Profit Calculation: Imagine a scenario where a company's profit (10d) is related to the number of units sold (d), expenses (-10d), and variable costs related to production (-8(-2d - 9)). Solving the equation helps determine the number of units needed to break even or achieve a specific profit level.

• Balancing Equations in Chemistry: While not directly analogous, the principles of combining like terms and isolating variables are crucial in balancing chemical equations. Stoichiometry, a vital part of chemistry, relies heavily on these algebraic skills. ScienceDirect publishes extensive work on chemical kinetics and stoichiometry.

• Physics problems: Concepts of forces, acceleration, etc. can be modeled through linear equations. Many introductory physics problems will involve solving similar equations to determine unknowns.

Conclusion

Solving the equation 10d = -10d - 8(-2d - 9) involves a series of straightforward algebraic steps. The solution, d = 18, is verified by substituting it back into the original equation. The underlying principles employed, however, extend far beyond this specific problem and are fundamental to numerous scientific and engineering disciplines. Understanding these principles provides a strong foundation for tackling more advanced mathematical concepts, as evidenced by the vast body of work on ScienceDirect exploring related topics. The practical applications demonstrate the equation's relevance in various fields, highlighting the significance of basic algebraic skills.