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When working with algebraic expressions, the substitution method is a fundamental technique used to simplify expressions or solve equations. This method involves replacing variables in an expression with their numerical values. For instance, if you're given an expression like \(5x - 4y\), and you know that \(x=7\) and \(y=5\), you apply the substitution method by replacing \(x\) with 7 and \(y\) with 5, resulting in \(5*7 - 4*5\).

It's essential to properly substitute values without altering the structure of the original expression. Here's a helpful tip: always use parentheses when substituting to prevent mistakes. For example, write \((5)(7) - (4)(5)\) to maintain clarity. Moreover, substitution not only helps in initial evaluations but also plays a crucial role in solving complex algebraic equations, where you might substitute one expression into another to find a solution.

The order of operations is a rule that defines the correct sequence in which to solve parts of a mathematical expression. This order is commonly remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Let's apply this to our expression \(5*7 - 4*5\). According to PEMDAS, we should carry out the multiplication before subtraction. Thus, we calculate \(5*7\) and \(4*5\) to get 35 and 20, respectively. Only after performing these multiplications do we proceed to subtract 20 from 35, which ultimately gives us the final answer, 15. Remember, failure to follow the order of operations can result in incorrect answers, so it's a critical step in evaluating algebraic expressions.

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols, but not equality or inequality signs. For example, \(5x - 4y\) is an algebraic expression, with \(x\) and \(y\) as variables. Understanding how to work with these expressions is vital for various areas of math, including algebra.

In an algebraic expression, each part has a name: '5x' is a term, '5' is the coefficient, 'x' is the variable, and '-4y' is another term composed of a coefficient and a variable. When evaluating an algebraic expression for particular values of its variables, like \(x=7\) and \(y=5\), you first substitute the values into the expression and then simplify the expression using the order of operations. Mastery of algebraic expressions lays the foundation for more complex topics in algebra, such as solving equations and inequalities.