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Understanding how to simplify expressions is fundamental when it comes to solving equations. Simplifying an expression means to make it easier to work with by combining like terms and using mathematical properties. For instance, in the given exercise, we simplify the expression by removing parentheses and understanding that subtracting a negative number is the same as adding the positive counterpart.

When we see an equation like \( x - (-9) = 15 \), it may look a bit complicated at first. However, simplification helps us see the equation more clearly. Recognizing the rule that two negatives make a positive, we rewrite the equation as \( x + 9 = 15 \). Such simplification steps make subsequent stages of solving equations much more straightforward.

Isolating the variable, often referred to as 'solving for x', is a crucial step in finding the solution to an equation. The goal is to get the variable on one side of the equation and all the other components on the other. This process usually involves performing 'inverse operations'.

For example, in our original problem, once we've simplified to get \( x + 9 = 15 \), we need to remove the 9 from the side with the variable, x. We do this by subtracting 9 from both sides of the equation because subtraction is the inverse of addition. The equation becomes \( x = 15 - 9 \), effectively isolating x on one side. It's like peeling away the layers of an onion to get to the center; here, x is our center.

The term 'equation operations' refers to the math actions we can perform on equations, such as addition, subtraction, multiplication, and division, as well as operations that involve exponents and roots when they are applicable. The principle behind using these operations is to maintain the balance of the equation by performing them equally on both sides.

In the original problem, after isolating the variable, we perform the operation of subtracting 9 from 15 to find the value of x. It's crucial to understand that whatever operation you perform on one side of the equation must also be performed on the other to keep the equation balanced. In this instance, we end up with \( x = 6 \), which is the solution to the equation. Remember, the goal of using equation operations is to simplify and ultimately solve the equation.