14b–2=–12b+4

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11-03-2025

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Solving the Equation 14b – 2 = –12b + 4: A Comprehensive Guide

This article explores the solution to the linear equation 14b – 2 = –12b + 4, providing a step-by-step process, discussing underlying mathematical concepts, and offering practical applications. We will delve into the solution method, explore potential variations, and examine the broader context of solving linear equations. This explanation avoids directly quoting ScienceDirect articles as the subject matter is elementary algebra, not requiring research articles for explanation. However, the structure and approach mimic the style of a scientifically-grounded explanation, emphasizing clarity and accuracy.

1. Understanding Linear Equations

Before diving into the solution, let's establish the fundamentals. A linear equation is an algebraic equation where the highest power of the variable (in this case, 'b') is 1. These equations represent straight lines when graphed on a coordinate plane. The goal when solving a linear equation is to isolate the variable, meaning to get the variable by itself on one side of the equation, thereby determining its value.

2. Solving the Equation: 14b – 2 = –12b + 4

Our primary objective is to find the value of 'b' that satisfies the equation. We'll achieve this by employing the following steps:

• Step 1: Combine like terms. We aim to group terms containing 'b' on one side of the equation and constant terms on the other. To do this, we'll add 12b to both sides and add 2 to both sides. This maintains the equality of the equation.

  14b – 2 + 12b + 2 = –12b + 4 + 12b + 2

  This simplifies to:

  26b = 6

• Step 2: Isolate the variable. Now, 'b' is multiplied by 26. To isolate 'b', we divide both sides of the equation by 26:

  26b / 26 = 6 / 26

  This simplifies to:

  b = 3/13

• Step 3: Verify the solution. It's crucial to check if the obtained value of 'b' satisfies the original equation. Substitute b = 3/13 into the original equation:

  14(3/13) – 2 = –12(3/13) + 4

  (42/13) – 2 = (–36/13) + 4

  Simplifying further:

  (42 – 26)/13 = (–36 + 52)/13

  16/13 = 16/13

Since both sides are equal, our solution b = 3/13 is correct.

3. Alternative Solution Methods:

While the above method is straightforward, alternative approaches exist. For instance, we could have initially subtracted 14b from both sides, leading to a slightly different sequence of steps but ultimately yielding the same result. The choice of method often depends on personal preference and the specific characteristics of the equation.

4. Applications of Linear Equations

Linear equations find widespread applications across various fields. Here are a few examples:

• Physics: Calculating velocity, acceleration, and displacement using kinematic equations, which are fundamentally linear relationships. For instance, distance = speed × time is a simple linear equation.

• Engineering: Designing structures, analyzing circuits, and modeling systems often involve solving systems of linear equations to determine optimal parameters or predict behavior.

• Economics: Linear equations are crucial in modeling supply and demand, forecasting economic growth, and analyzing market trends. For example, the relationship between price and quantity demanded might be represented by a linear equation.

• Computer Science: Linear algebra forms the foundation of many algorithms in computer graphics, machine learning, and data analysis. Solving linear equations is an integral part of these algorithms.

5. Expanding on the Concept: Systems of Linear Equations

The equation we solved is a single linear equation with one variable. However, many real-world problems require solving systems of linear equations with multiple variables. These systems can be solved using techniques like substitution, elimination, or matrix methods (Gaussian elimination, Cramer's rule).

For example, consider a system:

2x + y = 5 x – y = 1

This system can be solved using substitution or elimination to find the values of x and y that satisfy both equations simultaneously. Solving systems of linear equations is a more advanced topic but builds directly upon the fundamental skills used in solving single-variable equations.

6. Handling More Complex Equations:

While the equation 14b – 2 = –12b + 4 is relatively simple, the same principles apply to more complex linear equations. The key is always to isolate the variable through systematic application of arithmetic operations (addition, subtraction, multiplication, division), ensuring that the same operation is performed on both sides of the equation to maintain balance.

7. Conclusion:

Solving the linear equation 14b – 2 = –12b + 4 involves a systematic approach of combining like terms and isolating the variable. The solution, b = 3/13, is verified by substituting it back into the original equation. This seemingly straightforward problem illustrates the fundamental principles of solving linear equations, which are essential tools in various scientific and engineering disciplines. Understanding and mastering these principles forms a crucial foundation for tackling more complex mathematical problems in the future. The ability to manipulate and solve equations is a critical skill that underpins much of quantitative reasoning. Practicing different types of equations will improve proficiency and build confidence in mathematical problem-solving.