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Insolating the variable is one of the initial steps in solving equations. Here, the purpose is to gather the unknown variable on one side of the equation to determine its value easily. In our example, the variable is \(x\), and it is part of the term \(12x\). To isolate this term, adjust the equation by adding or subtracting numbers as needed.For instance, the equation \(12x - 15 = -3\) requires us to move \(-15\) away from the left. The method of choice is to do the opposite operation; since \(-15\) is subtracted, we add \(15\) to both sides. This step modifies the equation to \[12x - 15 + 15 = -3 + 15\] Which simplifies to \(12x = 12\). Once simplified, \(x\) is effectively isolated on the left side, making it straightforward to solve.

Simplifying equations is crucial because it makes them more manageable and easier to solve. By performing arithmetic operations, you can reduce complex expressions to simpler forms. In our example, simplifying occurred after isolating the variable term.By adding \(15\) to both sides, the equation became: \[12x = 12\] This step eliminated unnecessary numbers and brought us closer to finding \(x\). Remember, always perform the same operation on both sides of the equation to maintain its balance. Simplification helps ensure that the equation remains valid while we work towards the solution.Moreover, having fewer terms simplifies further steps, such as determining the final value of the variable. This efficient method streamlines the process, halting potential complications in the equation.

Algebraic manipulation is like solving a puzzle, adjusting the pieces until everything fits perfectly. In this exercise, manipulation involves rearranging terms and operations to unlock the value of \(x\). Once we've isolated the \(x\) in \(12x = 12\), our next step is to handle the operation binding the coefficient (12) to \(x\).To resolve this, divide both sides by 12: \[x = \frac{12}{12}\] In doing so, we perform algebraic manipulation to simplify further:- The coefficient \(12\) cancels out, removing itself as a hindrance to \(x\),- Resulting in \(x = 1\).This skill of manipulating equations helps in effectively solving them, showing how each adjustment unveils more of the path to the final answer. It involves clear-thinking and systematic approaches like dividing, factoring, or expanding, depending on what's needed.