n–3n+17=–1

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06-03-2025

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Solving the Equation: Unraveling the Mystery of n – 3n + 17 = –1

This article explores the seemingly simple algebraic equation, n – 3n + 17 = –1, breaking down the solution process step-by-step. We'll not only solve for 'n' but also delve into the underlying mathematical principles and consider the broader context of such equations within algebra. While this specific equation doesn't require advanced techniques, understanding its solution provides a strong foundation for tackling more complex problems. There's no direct referencing of ScienceDirect articles here, as this is a fundamental algebra problem not typically covered in research papers at that level. However, the principles applied are foundational to many mathematical concepts explored within higher-level mathematics and computational sciences.

1. Simplifying the Equation:

The first step in solving any algebraic equation is to simplify it as much as possible. Our equation, n – 3n + 17 = –1, contains like terms that can be combined.

• Like terms: These are terms that contain the same variable raised to the same power. In this case, 'n' and '-3n' are like terms.

Combining like terms, we get:

-2n + 17 = -1

This simplified equation is equivalent to the original but is easier to work with.

2. Isolating the Variable:

Our goal is to isolate the variable 'n' on one side of the equation. To do this, we need to perform inverse operations to eliminate the constants surrounding 'n'.

• Inverse operations: These are operations that undo each other (addition and subtraction, multiplication and division).

First, we subtract 17 from both sides of the equation:

-2n + 17 - 17 = -1 - 17

This simplifies to:

-2n = -18

3. Solving for 'n':

Finally, we need to solve for 'n' by dividing both sides of the equation by -2:

-2n / -2 = -18 / -2

This gives us the solution:

n = 9

4. Verifying the Solution:

It's always a good practice to verify the solution by substituting it back into the original equation:

9 – 3(9) + 17 = -1

9 – 27 + 17 = -1

-1 = -1

The equation holds true, confirming that our solution, n = 9, is correct.

5. The Broader Context: Linear Equations

The equation n – 3n + 17 = –1 is a type of linear equation. Linear equations are characterized by having a variable raised to the power of 1 (no exponents other than 1). They can be represented graphically as straight lines. The solution we found, n = 9, represents the x-intercept of the line if we were to graph the equation y = -2n + 17. The y-intercept would be at (0,17).

6. Practical Applications of Linear Equations:

Linear equations have countless applications in various fields:

• Physics: Calculating speed, distance, and time; modeling projectile motion.
• Engineering: Designing structures, analyzing circuits, and controlling systems.
• Economics: Modeling supply and demand, predicting economic growth.
• Computer Science: Developing algorithms, creating simulations, and solving optimization problems.

For example, consider a scenario where a company's profit (P) is given by the equation P = 10x - 50, where 'x' represents the number of units sold. To find the break-even point (where profit is zero), we set P = 0 and solve for x:

0 = 10x - 50

10x = 50

x = 5

This means the company needs to sell 5 units to break even. This is a simple application of a linear equation, illustrating their practicality in real-world situations.

7. Extending the Concept: More Complex Equations

While our initial equation was relatively straightforward, the principles we used—simplifying, isolating the variable, and verifying the solution—are applicable to more complex algebraic equations. For instance, consider the quadratic equation:

x² + 5x + 6 = 0

While this equation involves a squared variable, the core methodology remains similar: factoring, the quadratic formula, or completing the square can be used to find the solutions for 'x'.

8. Conclusion:

Solving the simple equation n – 3n + 17 = –1 provides a valuable stepping stone in understanding algebraic manipulation. The process, encompassing simplification, isolating the variable, and verifying the solution, forms the bedrock of solving more complex algebraic problems. The concepts discussed here extend far beyond simple equations, underpinning numerous applications in various fields of science and engineering. Understanding linear equations and their solution methods is a crucial skill for anyone pursuing studies or a career in quantitative fields. By mastering these fundamentals, you pave the way for tackling more advanced mathematical concepts with confidence.