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Solving equations is a fundamental skill in mathematics and is all about finding the value of the unknown variable that makes the equation true. Think of equations as a balance scale, where both sides must remain equal. In this process, we perform operations on both sides of the equation to keep the balance intact, while simplifying to discover the value of the variable.
To solve the equation \(-4.2x = 50.4\), we aim to identify which operations will 'undo' the actions taken on the variable. By reversing these steps, we can isolate the variable on one side of the equation. This requires familiar operations like addition, subtraction, multiplication, and division, depending on how the variable is related to constants and coefficients in the equation.

Division is a powerful tool when working with equations, particularly when you need to eliminate a coefficient attached to a variable. When the variable is multiplied by a number, division helps in 'undoing' this multiplication. This is based on the principle of the multiplicative inverse, which states that dividing by a number is the same as multiplying by its reciprocal.
In our example equation \(-4.2x = 50.4\), we see that \(x\) is multiplied by \(-4.2\). To solve for \(x\), we perform division on both sides by \(-4.2\). Remember, whatever you do to one side of the equation, you must do to the other to maintain equality. This principle is crucial, as it keeps the equation balanced and allows us to solve for the unknown correctly.

Isolating variables is the goal of solving equations, as it provides the solution to the problem. The process involves manipulating the equation such that the variable of interest stands alone on one side. This is achieved by systematically undoing any mathematical operations using inverse operations—each step is about stripping away layers until the variable is isolated.
In the example, \(-4.2x = 50.4\), we identify that the multiplication by \(-4.2\) is the operation surrounding \(x\). We choose division (the inverse operation of multiplication) to cancel out this effect, thus isolating \(x\). When simplified, this gives us \(x = -12\), thereby demonstrating that \(x\) indeed stands alone, revealing the solution. Remember, isolating the variable is akin to peeling an onion, tactfully removing layers until only the kernel remains.