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When solving linear equations, the first and one of the most critical steps is to 'isolate the variable.' This means that we want to have the variable (in this case, 'x') by itself on one side of the equation. To achieve this, we need to perform operations that will eliminate any coefficients or numbers attached to the variable.

Think of it like balancing a scale. If there is a number tied to the variable side, you have to do something to remove that number and keep the scale balanced. In our exercise, the equation \[\begin{equation}-12x = 3\end{equation}\] presents us with a variable, 'x,' that is multiplied by -12. To isolate 'x,' we have to get rid of this -12.

The most straightforward way to do this is through division - by dividing both sides of the equation by -12, we are able to 'free' the variable. This results in the equation \[\begin{equation}x = \frac{3}{-12}\end{equation}\], which forms our next step towards finding the value of 'x.'

Once we have isolated the variable, our next task is to 'simplify the equation.' This step makes the answer more understandable and easier to work with, especially when dealing with fractions or large numbers.

The goal of simplification is to clean up the equation and make the solution as clear as possible. In the example \[\begin{equation}x = \frac{3}{-12}\end{equation}\], we have a fraction that can be reduced to lowest terms, making our understanding and computation of the value simpler. The numerator and denominator in our fraction have a common factor that we can divide by - in this case, 3.

When we do so, the equation simplifies to \[\begin{equation}x = -0.25\end{equation}\] Now, 'x' is presented in a simplified decimal form, which is more intuitive to most students.

Checking the solution is an essential habit to cultivate when solving any mathematical problem. This step ensures that our answer is correct and reinforces our understanding of the principles applied.

In our example, we arrived at \[\begin{equation}x = -0.25\end{equation}\] as the solution. To verify this, we substitute '-0.25' back into the original equation \[\begin{equation}-12x = 3\end{equation}\] to see if the left side equals the right side. This process can be thought of as reversing our steps.

Through substitution, we get \[\begin{equation}-12(-0.25) = 3\end{equation}\], which simplifies to 3 = 3, verifying that our solution is indeed correct. Always perform a check after solving, as it assures you that the variable was correctly isolated and the equation simplified properly, leaving no room for errors.