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 explores the solution to the algebraic equation 10d = -10d - 8(-2d - 9). We'll break down the process step-by-step, explaining the underlying principles of algebra and offering practical examples to solidify your understanding. While this specific equation doesn't appear directly in a readily accessible ScienceDirect article (as it's a relatively simple linear equation), we will approach its solution using standard algebraic techniques, referencing general principles found in numerous algebra textbooks and resources available on ScienceDirect and elsewhere that cover similar topics.

Understanding the Equation

The equation 10d = -10d - 8(-2d - 9) is a linear equation in one variable (d). This means the highest power of the variable 'd' is 1. Solving the equation involves isolating the variable 'd' on one side of the equation to find its value.

Step-by-Step Solution

1. Distribute the -8: The first step is to distribute the -8 to the 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: Subtract 6d from both sides of the equation to get all the 'd' terms on one side:

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

   This simplifies to:

   4d = 72

4. Solve for d: Divide both sides of the equation 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 – Relevance to Science and Engineering

While this particular equation might seem simple, the underlying principles are fundamental to various scientific and engineering applications. Solving linear equations is crucial in:

• Physics: Many physics problems, particularly in mechanics and electricity, involve solving linear equations to determine unknown quantities like velocity, acceleration, force, or current. For example, Ohm's Law (V = IR) is a simple linear equation relating voltage (V), current (I), and resistance (R).

• Chemistry: Stoichiometry, the study of quantitative relationships in chemical reactions, relies heavily on solving linear equations to determine the amounts of reactants and products involved.

• Engineering: Engineers frequently use linear equations to model and analyze systems. For example, calculating stress and strain in a material under load, analyzing circuits, and designing control systems all involve solving systems of linear equations.

• Data Analysis and Modeling: In many scientific fields, linear equations are used to create simple models of relationships between variables. For example, a linear regression model fits a straight line to data points, which is essentially solving a linear equation to find the best-fitting line. (Note that more complex models exist, but the foundation often involves solving linear equations).

Further Applications and Extensions

The techniques used to solve this simple linear equation can be extended to solve more complex problems:

• Systems of linear equations: Many real-world problems require solving multiple linear equations simultaneously. Techniques like substitution, elimination, or matrix methods are used to find solutions.

• Nonlinear equations: Equations with higher powers of the variable (quadratic, cubic, etc.) require different solution methods, often involving factoring, the quadratic formula, or numerical techniques.

• Differential equations: These equations involve derivatives and are essential in modeling dynamic systems in physics, engineering, and biology.

Conclusion

Solving the equation 10d = -10d - 8(-2d - 9) demonstrates fundamental algebraic techniques. While seemingly straightforward, the principles involved are vital for tackling more complex problems across numerous scientific and engineering disciplines. Mastering these foundational skills is crucial for anyone pursuing studies in STEM fields. By understanding the systematic approach to solving linear equations, you build a strong base for more advanced mathematical concepts. Remember to always verify your solutions to ensure accuracy and develop a solid understanding of the underlying principles.