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When working with algebraic expressions, the substitution method is a cornerstone technique. It involves replacing variables with their corresponding numerical values. For example, when given an expression like \(5x - 4y\) and asked to evaluate it for \(x=7\) and \(y=5\), what you're essentially being asked to do is to replace the \(x\) with 7 and the \(y\) with 5. This method turns an abstract algebraic expression into a more concrete arithmetic problem.

Let's look closer at how this is done. Initially, the expression holds variables: \(5x - 4y\). After applying the substitution method, these variables are switched out, leading to \(5(7) - 4(5)\), a much simpler equation to handle since it involves only numbers and basic operations. Remember, before moving on to the operations, make sure all variables have been substituted properly to avoid any errors. This method offers a clear pathway from a general expression to a specific numerical answer.

Once the substitution has been completed, it's time to simplify the expression. Simplifying is all about combining and reducing the components of an algebraic expression to its simplest form using arithmetic operations. Often, it involves performing multiplication, division, and handling like terms. In our example \(5(7) - 4(5)\), we simplify by first calculating the multiplication.

How does this work exactly? We multiply the numbers directly, transforming \(5(7)\) into \(35\) and \(4(5)\) into \(20\). The expression is now much simpler: \(35 - 20\). Note that simplification is a step-by-step process: deal with parentheses and exponents first, then multiplication and division, and finally addition and subtraction. The goal is a single number or a less complex expression. For students, learning to recognize the order in which to tackle different operations is key to mastering simplification.

With the expression simplified to \(35 - 20\), the final step in evaluating the expression is to perform the arithmetic operations. Arithmetic is the branch of mathematics that deals with numbers and operations on them, such as addition, subtraction, multiplication, and division. These basic operations are the building blocks for more complex mathematical work, and they follow a particular order known as the order of operations.

In our case, we only need to subtract, which is straightforward: we simply take 20 away from 35, leaving us with 15. Arithmetic operations are the final strokes in solving numerical expressions. It's important to perform these operations correctly to arrive at the correct solution, and not to rush, even when the calculations seem simple. Attention to detail and following the correct sequence can make a big difference to the accuracy of your result.