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In algebra, substituting variables forms the cornerstone of solving expressions and equations. It involves replacing a letter or symbol with its given numerical value. This is critical because it allows you to work with concrete numbers rather than abstract symbols. For instance, in the expression \(13+3b\), the first step to evaluate it for \(b=7\) is to replace \(b\) with the number 7, resulting in a new expression \(13+3*7\).

When performing substitution, it is important to ensure that every instance of the variable is replaced. To improve comprehension, visualize the variable as a placeholder for the number it represents. It's akin to putting a picture into a picture frame—once the variable is replaced with its value, the overall look (or the expression in this case) changes. Through this process, an abstract algebraic phrase is transformed into a straightforward arithmetic problem.

The order of operations is essential in mathematics to avoid confusion and to ensure consistency in the evaluation of expressions. You might recall the acronym 'PEMDAS' or 'Please Excuse My Dear Aunt Sally', representing the sequence in which operations should be applied: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In the expression \(13+3*7\), we must follow PEMDAS. Thus, multiplication should be performed before addition. After substituting \(b=7\), we multiply \(3*7\) to obtain 21, and only then do we proceed to add the result to 13. By faithfully following the order of operations, you can accurately solve complex expressions without ambiguity. Always remember that deviating from this order can lead to incorrect results, much like baking a cake without following the recipe will lead to a dessert that doesn't rise properly.

The term arithmetic operations encompasses the basic building blocks of math: addition, subtraction, multiplication, and division. These operations allow us to perform a vast array of mathematical tasks, from the simplest to the most complicated ones. To solidify understanding, let's focus on addition and multiplication, as seen in the provided example.

After substituting \(b=7\) and following the order of operations, we are left with two arithmetic tasks: \(3*7\) and then \(13+21\). Multiplication is repeated addition, so you can think of \(3*7\) as adding three sevens together. After obtaining 21, we add it to 13, a direct and straightforward arithmetic operation. Mastery of these basic functions is necessary not only to solve algebraic expressions but also as the foundation for learning more advanced mathematical concepts. Remember, practice makes perfect, so continue working through various problems to become adept at these operations.