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The substitution method is one of the simplest ways to evaluate algebraic expressions. It involves replacing a variable in an expression with a given value to simplify the equation. This method helps to evaluate expressions easily without any need for guesswork. Here’s how it works:

• Identify the variable in the expression, in this case, it's any given value of \(x\).
• Replace each occurrence of this variable in the expression with the input number.
• Simplify the expression using arithmetic operations.

For example, in the expression given: \[9 \times 2 - 1 \times x - 7\], if \(x = 0\), substituting this value into the expression simplifies it to: \[9 \times 2 - 1 \times 0 - 7 = 18 - 0 - 7 = 11\]. By methodically substituting each \(x\)-value, you can evaluate each specific situation.

A variable expression is an algebraic expression that includes both numbers and variables, like the expression \[9 \times 2 - 1 \times x - 7\]. Here, \(x\) is the variable. The role of the variable is to act as a placeholder for any number that can be substituted into the expression.
The expression changes its value based on what number replaces \(x\).
This flexibility is what makes algebraic expressions very useful for representing mathematical relationships.

• Variables allow us to model real-world situations where quantities can vary.
• They enable generalization in solving equations.

So, when encountering a variable expression, remember that these expressions can represent many numbers. By evaluating them for different values of the variable, you can see how the equation behaves under various conditions.

Order of operations is crucial when simplifying expressions correctly. It ensures all calculations are done in an agreed order which makes the results reliable and consistent. Remember the acronym PEMDAS:

• Parentheses - Solve expressions inside parentheses first.
• Exponents - Next, evaluate all powers/exponents.
• Multiplication and Division - From left to right, do all multiplication and division.
• Addition and Subtraction - Finally, handle addition and subtraction from left to right.

In our scenario: after substituting, you follow these steps to simplify. For instance, \(9 \times 2 - 1 \times 0 - 7\) was evaluated by first multiplying: \(9 \times 2\) to get 18, then handled subtraction following the order of operations. Ensuring you execute operations in this order prevents mistakes and produces correct outcomes every time.