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Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. For instance, in the expression \(\frac{x-5}{6}=12-x\), \(x\) represents a variable, which can be any number. When we are given an equation involving an algebraic expression, our goal is to find the value of the variable that makes the equation true. This often requires us to perform a series of operations to manipulate these expressions.

In our example, we begin with \(\frac{x-5}{6}\) where \(x\) is multiplied by \(\frac{1}{6}\) after subtracting 5. Understanding how to work with these operations is critical in algebra. It allows us to solve equations systematically and to determine if a certain number is a solution to the equation.

Equation simplification is about making complex expressions more manageable by reducing them to simpler forms. This process usually involves combining like terms, reducing fractions, and performing arithmetic operations. For any equation, simplification is a crucial step that often precedes the actual solving process.

In the given exercise, simplifying \(\frac{11-5}{6}\) to 1 is a prime example of equation simplification. Similarly, reducing \(12 - 11\) to 1 makes it easier to compare both sides of the equation. Simplification clarifies the relationship between the variables and constants involved and provides a clearer path to the solution. It also reduces the likelihood of making arithmetic mistakes in the subsequent solving steps.

The substitution method is particularly useful when you have an equation with an unknown variable and you want to check if a particular value is a solution. It involves replacing the variable with the given number and assessing whether the left-hand side of the equation equals the right-hand side.

For our exercise, the substitution method is straightforward. We take the number 11, which is our potential solution, and substitute it in place of \(x\) in the original equation. After simplifying both sides, if we find that both sides of the equation are equal - as with our 1 = 1 result - we can confidently state that 11 is a solution to the equation. This method is powerful because it can be used not just for simple equations but also for more complex systems where multiple substitutions may be necessary to find a solution.