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Understanding inverse operations is crucial when it comes to solving linear equations like the given example \( -9 = 5 + x \). An inverse operation reverses the effect of the original operation performed on a variable. In simple terms, it's the 'undo' button for math operations. For instance, if you have added a number, the inverse operation is to subtract that same number.

In our example, the variable \(x\) is being increased by 5. To isolate \(x\) and solve for it, we do the opposite: we subtract 5 from both sides of the equation. It's like saying we're removing what was added to get back to the original value of \(x\). These operations are precisely what makes solving equations possible because they allow us to simplify complex problems into a solvable form.

The process of equation solving can be broken down into systematic steps that, when followed carefully, lead to the correct solution of the equation. It's like following a recipe to make sure your cake comes out perfect every time.

For our linear equation \( -9 = 5 + x \), the steps can be outlined as follows:

• Identify what you want to find: in this case, the value of \(x\).
• Determine the inverse operation: here, since \(x\) is increased by 5, we will subtract 5.
• Apply the inverse operation to both sides: we get \( -9 - 5 = 5 - 5 + x \), which simplifies to \( -14 = x \).

These steps provide a structured approach to solving equations, making it more manageable for students to navigate through any math problem.

After solving an equation, it's always good practice to check your solution to ensure its accuracy. Just like proofreading your essay before submitting it, this step confirms that your answer actually works in the original equation.

For our problem, we've found that \(x = -14\). To check this, we substitute \( -14 \) back into the initial equation in place of \(x\). If both sides of the equation balance, meaning they are equal, then we can confidently say our solution is correct. So, we calculate \( 5 + (-14) \), and it indeed equals \( -9 \), which matches the left side of our equation. This confirms that \( -14 \) is the correct solution for \(x\).