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The substitution method in algebra is a fundamental technique used to evaluate expressions with variables. When an expression includes variables for which values are provided, we can replace these variables with their corresponding values. This process allows us to transform an algebraic expression into a numeric one, which can then be simplified and evaluated.

Consider evaluating \(\frac{x_{1}+x_{2}}{2}\) for given values. Here's how it works:

• Identify the variables in the expression, which are \(x_{1}\) and \(x_{2}\) in this case.
• Replace each variable with the given value: if \(x_{1}=4\) and \(x_{2}=-7\), substitute to get \(\frac{4+(-7)}{2}\).
• The expression no longer contains variables and is ready for further simplification.

This strategy is useful not only for simple expressions but also for solving equations and inequalities, making it a backbone of algebraic manipulation.

Once the substitution method has been applied and the variables are replaced with actual numbers, the focus shifts to simplifying numerical expressions. Simplification involves carrying out the operations of addition, subtraction, multiplication, and division, in accordance with the order of operations (PEMDAS/BODMAS). Applied correctly, simplification reduces an expression to its simplest form.

For our example, after substitution, we get \(\frac{4+(-7)}{2}\). The next step is to combine the integers in the numerator. The expression simplifies further as \(4-7=-3\), leaving the simplified expression of \(\frac{-3}{2}\). It's essential to always combine like terms and execute operations correctly to ensure the expression simplifies accurately. Proper simplification leads to accurate evaluations and is a skill that enhances algebraic fluency.

Arithmetic with integers incorporates working with positive and negative whole numbers. In algebra, it's often intertwined with simplifying expressions and solving equations. A sound understanding of integer rules is imperative.

When simplifying expressions such as \(\frac{4+(-7)}{2}\), recognizing the operation needed between two integers is crucial. Here, you're combining a positive integer, 4, with a negative integer, -7. Remember that adding a negative is the same as subtracting the positive, hence 4 add -7 becomes 4 subtract 7.

Following this, you handle the division by 2. As \(\frac{-3}{2}\) is not divisible evenly, it results in a non-integer, which is -1.5 in decimal form. This step shows how integer rules extend beyond whole numbers and into rational numbers, ensuring a firm grasp of arithmetic extends the understanding of algebraic concepts.