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Solving algebraic equations involves finding the value or values of the variable that make the equation true. In our example, we started with the equation \(3 \times 2 = x + 13 - x\). An algebraic equation is a statement that two expressions are equal, and our goal is to determine what value of \(x\) will satisfy this condition. To solve an algebraic equation, typically we want to isolate the variable on one side of the equation. This allows us to clearly see what value the variable must take. Solving equations often involves using operations like addition, subtraction, multiplication, and division to simplify and manipulate the expression.

Algebraic simplification is about making an equation easier to understand and work with by reducing it to its simplest form. In the exercise given, we first simplify both sides of the equation. Bringing complex expressions to simpler terms makes them more manageable. For example, on the left side of \(3 \times 2 = x + 13 - x\), we compute \(3 \times 2\) to get 6. On the right side, the terms \(x\) and \(-x\) cancel each other out. Thus, what remains is 13.

The ultimate aim of simplification is to reach an expression where no further simplification is possible, and to help us see more clearly what is happening in the equation. The simplification process is essential in solving equations as it lays out the path to finding the solution or determining that no solution exists.

Equation solving involves a systematic approach to finding solutions. The steps we followed began with simplifying both sides of our equation \(3 \times 2 = x + 13 - x\). By simplifying, we found that both sides of the equation differ (6 = 13), indicating no solution exists.

Here are the general steps for any equation solving task:

• Identify the parts of the equation and simplify each side.
• Use mathematical operations to isolate the variable on one side.
• Assess whether both sides are equal to determine if a solution exists.

In this scenario, rather than finding a value for \(x\), simplification showed that the equation itself was not valid. A key part of solving algebraic equations is understanding that sometimes the equation may not have a solution. Recognizing such cases is as crucial as solving equations that do have solutions.