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An algebraic expression is a combination of numbers, variables (like x or y), and mathematical operations (such as addition, subtraction, multiplication, and division). For example, in the expression \(5x - 7\), \(5x\) is a term where 5 is the coefficient and x is the variable. Understanding algebraic expressions is crucial because they allow us to generalize arithmetic operations and solve equations more easily. When working with algebraic expressions, you may often need to simplify them or perform operations such as addition, subtraction, and particularly multiplication, which we'll dive into next.

Multiplication is one of the basic operations in mathematics, and it can be applied to both numbers and algebraic expressions. For multiplication involving algebraic expressions, like in our original exercise with the distributive property, it's essential to multiply each term inside the parentheses by the term outside.

For example, if you have \(a(b - c)\), you distribute a to both b and c, resulting in \(ab - ac\). In our given problem: \(-\frac{1}{5}(5x - 7)\), we distribute \(-\frac{1}{5}\) to both \(5x\) and \(-7\) separately:

\(-\frac{1}{5} \times 5x\) and \(-\frac{1}{5} \times (-7)\).

This concept is important not just for solving this particular problem, but for understanding algebra as a whole.

Negative numbers are numbers less than zero, and they often appear in algebraic expressions. When multiplying negative numbers, there are a few rules that are important to remember.

First, when you multiply a negative number by a positive number, the result is always negative. However, when you multiply two negative numbers, the result is positive.

In our exercise, we see this with: \(-\frac{1}{5}\) multiplied by \(-7\), which results in a positive \frac{7}{5}\ because multiplying two negatives gives a positive result.

Understanding how negative numbers work is essential for mastering algebraic operations, as it ensures accurate and efficient problem solving in more complex equations.