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The substitution method plays a crucial role in mathematics, particularly in evaluating expressions. It involves replacing variables with their given numerical values.

Take, for example, the expression \(5(x-4)\) when \(x=-3\). The first step is to substitute \(x\) with \( -3\), resulting in an updated expression: \(5(-3-4)\). This is a straightforward process, but care must be taken to substitute the variable correctly to avoid errors in subsequent steps.

• Identify the variable(s) in the expression.
• Determine the value(s) that will be substituted for the variable(s).
• Replace the variable(s) with the given number(s).

In our example, the variable is \(x\) and the number we are substituting is \( -3\). By correctly replacing \(x\) with \( -3\), we lay a solid groundwork for correctly simplifying the rest of the expression.

Once you've completed the substitution step, the next phase is simplifying expressions. Simplification makes an expression easier to understand and work with by reducing it to its simplest form.

In the expression \(5(-3-4)\), simplification occurs in two stages:

Stage 1: Combine Like Terms

First, address any arithmetic within the parentheses. \( -3\) and \( -4\) are like terms, as they are both numbers without variables, so we combine them to get \( -7\). This is efficient and prevents mix-ups in later stages.

Stage 2: Multiply

With the terms combined, multiply the resulting number by 5, as per the original expression \(5(-7)\), resulting in \( -35\).

• Combine like terms, if any are present.
• Perform any arithmetic operations according to the order of operations.
• Reduce the expression to as simple a form as possible.

When simplified, expressions are more concise and manageable, which is why this step is an important skill in algebra and beyond.

A foundation for correctly simplifying expressions lies in the order of operations, often remembered by the acronym PEMDAS or BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

This hierarchy determines which parts of an expression to simplify first. In the given example, \(5(-3-4)\), we tackle the parentheses first, as they take precedence. Within the parentheses, we perform the subtraction (\( -3\) minus \( -4\)), yielding \( -7\). Following the order of operations, multiplication comes next, so we then multiply \(5\) by \( -7\) to get \( -35\).

Without following this order, we risk arriving at incorrect answers. For instance, if we multiplied before subtracting, we would end up with an entirely different and incorrect result. Always remember, respecting the order of operations is key to correct calculation in algebra and throughout mathematics.