n–3n+17= – 1

Buzhidao authors

3 min read

·

06-03-2025

4

0

Share

Solving the Equation: n – 3n + 17 = –1 – A Deep Dive into Linear Equations

This article explores the solution to the linear equation n – 3n + 17 = –1. While seemingly simple, this equation provides a valuable opportunity to understand the fundamental principles of algebra, particularly how to manipulate equations to isolate the unknown variable (in this case, 'n'). We will break down the solution step-by-step and then delve into broader applications and related concepts. No external sources like ScienceDirect are needed for this specific equation, as its solution is a fundamental algebraic concept.

1. Simplifying the Equation:

The first step in solving any equation is to simplify it. Our equation, n – 3n + 17 = –1, contains like terms that can be combined. The terms 'n' and '-3n' are both multiples of 'n'. Combining them, we get:

-2n + 17 = -1

This simplified equation is equivalent to the original; we've just made it more manageable.

2. Isolating the Variable:

Our goal is to isolate 'n' on one side of the equation. To do this, we need to perform inverse operations. Currently, '17' is added to '-2n'. The inverse operation of addition is subtraction. We subtract 17 from both sides of the equation to maintain balance:

-2n + 17 – 17 = -1 – 17

This simplifies to:

-2n = -18

3. Solving for 'n':

Now, 'n' is multiplied by -2. The inverse operation of multiplication is division. We divide both sides of the equation by -2:

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

This gives us the solution:

n = 9

Therefore, the solution to the equation n – 3n + 17 = –1 is n = 9.

4. Verification:

It's always good practice to verify our solution. We substitute n = 9 back into the original equation:

9 – 3(9) + 17 = -1

9 – 27 + 17 = -1

-18 + 17 = -1

-1 = -1

The equation holds true, confirming our solution is correct.

5. Expanding the Concept: Linear Equations and Their Applications:

This simple equation exemplifies a linear equation—an equation where the highest power of the variable is 1. Linear equations are fundamental in mathematics and have widespread applications in various fields:

• Physics: Describing motion (e.g., calculating velocity or distance), analyzing circuits, and modeling simple harmonic motion often involve linear equations. For example, calculating the final velocity of an object given its initial velocity, acceleration and time utilizes a linear equation.

• Engineering: Linear equations are used extensively in structural analysis, fluid mechanics, and control systems. They help engineers model and predict the behavior of systems under different conditions. For example, calculating the load-bearing capacity of a beam might involve solving a system of linear equations.

• Economics: Linear equations can model supply and demand curves, helping economists predict market behavior. For example, a simple supply curve might be expressed as P = aQ + b, where P is the price, Q is the quantity, and a and b are constants.

• Computer Science: Linear equations are crucial in computer graphics, image processing, and machine learning algorithms. Many computer graphics transformations are linear and solving systems of linear equations is frequently needed for efficient computations.

• Everyday Life: We implicitly use linear equations in everyday situations. For instance, calculating the total cost of items given their individual prices, determining the distance traveled based on speed and time, or figuring out how much paint to buy based on the area to be covered all involve the principles of linear equations.

6. Solving More Complex Linear Equations:

While our example was straightforward, linear equations can become more complex. They might involve:

• Fractions: Equations containing fractions require finding a common denominator to simplify before proceeding with the solution process.
• Decimals: Equations with decimals can be solved directly or by multiplying the entire equation by a power of 10 to eliminate the decimals.
• Multiple Variables: Systems of linear equations involving multiple variables require techniques like substitution or elimination to find the values of all variables. Matrix methods are also efficient for solving large systems of linear equations.

7. Conclusion:

Solving the equation n – 3n + 17 = –1 provides a foundational understanding of solving linear equations. The step-by-step process – simplifying, isolating the variable, and verifying the solution – is applicable to a wide range of algebraic problems. The widespread application of linear equations in diverse fields highlights their importance as a fundamental mathematical tool. Understanding and mastering the solution of these equations is crucial for success in many academic and professional pursuits. Further exploration of solving systems of linear equations, and more advanced equation types, will build upon this foundational understanding.