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Isolating variables is a crucial step in solving linear equations. The idea is to get the variable, often represented by letters like \( x \), on one side of the equation by itself. This makes finding the value of the variable much simpler. Think of it as tidying up a room: move everything else out of the way to focus on one item.
For the equation \(-3 - 4x = -47\), the first step is to eliminate the constant term \(-3\) from the left-hand side. You can do this by performing the inverse operation: adding \( 3 \) to both sides.

• This leaves us with \(-4x = -44\), effectively isolating the \(-4x\).

Once you have the variable term isolated, you're one step closer to solving the equation. It's important to approach this systematically to avoid any potential errors.

Inverse operations are operations that "undo" each other. They are key to solving equations because they allow you to simplify an equation progressively by reversing operations performed on the variable.
The basic inverse operations involve:

• Addition and subtraction
• Multiplication and division

In our example of \(-3 - 4x = -47\), we used addition as an inverse operation to remove the \(-3\):

• By adding \( 3 \) to both sides, we cancel out \(-3\), leaving us with \(-4x = -44\).
• The next step is to use division, the inverse of multiplication, to solve for \( x \). By dividing both sides by \(-4\), we find \( x = 11\).

Inverse operations help maintain the equality of the equation while simplifying it to find the solution easily.

Using step-by-step solutions when solving equations ensures a clear and methodical approach. It helps break down complex problems into manageable parts, making them easier to solve. Here’s how we applied step-by-step logic to the equation \(-3 - 4x = -47\): First, we isolated the variable term. By adding \( 3 \) to both sides, we clarified the equation structure:

• The result was \(-4x = -44\).

Then, we solved for \( x \) by performing the inverse operation of multiplication, which is division. We divided each side by \(-4\), yielding:

• \( x = 11\), which is the final solution.

This systematic method reduces confusion and highlights any errors immediately. Following a step-by-step solution is like following a recipe; each action leads naturally to the next until the result is achieved.