isolate

In mathematical equations, the procedure of singleing out a specific variable on one side of the equation is a fundamental algebraic technique. This involves performing operations on both sides of the equation to leave the desired variable by itself, thereby revealing its value in terms of other constants or variables. For instance, given the equation 2x + 3 = 7, the variable ‘x’ is singled out by subtracting 3 from both sides, resulting in 2x = 4, and then dividing both sides by 2, leading to x = 2.

This manipulation is crucial for solving equations, understanding relationships between variables, and simplifying complex expressions. Its widespread use stems from its ability to transform seemingly intractable problems into manageable forms. Historically, the development of algebra and symbolic manipulation has relied heavily on this process, enabling advancements in various fields such as physics, engineering, and economics.

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In mathematics, to isolate a variable or mathematical expression signifies the process of manipulating an equation or inequality to have that specific variable or expression alone on one side, typically the left side. This is accomplished through the strategic application of inverse operations to both sides of the equation, maintaining equality while progressively simplifying the expression. For instance, in the equation x + 3 = 7, the act involves subtracting 3 from both sides to obtain x = 4, thereby achieving the aim.

The significance of this procedural step resides in its fundamental role in problem-solving across various mathematical domains. It enables the determination of the value of an unknown quantity and is pivotal in simplifying complex equations for further analysis. Historically, the development of algebraic manipulation techniques, including the concept, facilitated advancements in fields such as physics, engineering, and economics, where mathematical models are extensively employed.

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