91Ó°ÊÓ

Algebra is a fascinating branch of mathematics that uses symbols and letters to represent numbers and quantities in equations and formulas. This makes it easier to generalize problems and create mathematical models when faced with real-world scenarios or abstract concepts.

The primary purpose of algebra is to find the unknown values that satisfy given relationships expressed in equations. For example, in our exercise, the goal is to solve for the unknown variable \(x\) in the equation:

• \(-6x - 14 = 4\)

To make sense out of this equation, we leverage certain algebraic principles that allow us to manipulate and transform the equation into a simpler form. These principles are crucial for understanding how different parts of the equation relate to one another, setting the foundation for finding solutions.

The phrase "isolation of variables" refers to the algebraic process of simplifying an equation so that the unknown variable is by itself on one side of the equation. This enables us to directly find its value.

In the given exercise, we start by identifying \(x\) as the variable we need to isolate. Initially, \(x\) is part of a larger expression, \(-6x - 14 = 4\). Our first task in isolating \(x\) is to remove the constant term, \(-14\), which is done by adding 14 to both sides. This eliminates the constant on the left and yields a new equation with just the variable term:

• \(-6x = 18\)

Next, to completely isolate \(x\), we need to eliminate the coefficient \(-6\) that is multiplied with \(x\). This is achieved by dividing both sides of the equation by \(-6\), giving us the solution:

• \(x = -3\)

Each step in this process systematically moves us closer to having \(x\) alone, allowing us to readily find its value.

A step-by-step solution is essential for solving linear equations methodically and reducing errors. It provides a clear and structured path to arrive at the correct solution.

The given problem: \(-6x - 14 = 4\), was approached in two key steps:1. **Addition/Subtraction Step:** Add 14 to both sides to remove the constant and streamline the equation. This simplification leaves us with:

• \(-6x = 18\)

2. **Multiplication/Division Step:** Divide both sides by \(-6\) to isolate \(x\). Through this operation, we determine:

• \(x = -3\)

Breaking down the problem into these manageable parts not only simplifies the process but also makes it easier to verify each transformation leading to the answer. This structured approach is integral to mastering algebra and solving even more complex equations.