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Solving equations is a foundational skill in algebra. It involves finding the value of the unknown variable that makes the equation true. In this exercise, we start by translating a word problem into an algebraic equation, which is often the first step in solving equations.

An algebraic equation is a mathematical statement where two expressions are set equal to each other. Here, our equation is \( x + 36 = -12 \). To solve this equation, we need to perform operations that help isolate the variable \( x \).

By performing the same operation on both sides of the equation, we maintain the balance of the equation. This means subtracting 36 from both sides in this instance, resulting in \( x = -48 \). This ensures that the equation remains valid and leads us to the correct solution.

When working with equations, unknown variables represent values we need to find. In our example, the variable \( x \) stands for the number we need to determine in order to make the equation true.

Variables act as placeholders for numbers we don't yet know but can calculate using the rules of algebra. They are crucial because they allow us to set up equations that model real-world situations and solve various kinds of problems.

Understanding variables involves recognizing how they interact within the equation to drive us towards the solution. By manipulating the other terms around them using algebraic operations, we eventually reveal their value.

Basic arithmetic operations are the building blocks of solving algebraic equations. They include addition, subtraction, multiplication, and division.

In our example, subtraction was used to isolate the variable \( x \). By subtracting 36 from both sides of the equation \( x + 36 = -12 \), we simplify the equation to \( x = -48 \).

Using arithmetic operations correctly is key to solving equations effectively. While each operation performs a different function, they all serve the purpose of simplifying equations and making it possible to solve for the unknown. With practice, understanding and applying these operations becomes second nature.