Problem 77 - Introductory Algebra for College Students

Question

Simplify each algebraic expression. $$-5(2 x)$$

Step-by-Step Solution

Answer

The simplified form of the algebraic expression is \(-10x\).

Step 1: Distribute the negative sign

When we have \(-5(2x)\), the negative sign should be distributed to the term next to it, which is \(2x\). After multiplying \(-5\) into \(2x\), we get \(-10x\).

Step 2: Simplify

In this case, there are no other terms to combine with \(-10x\), so the expression cannot be simplified any further. Therefore, the simplified form of the expression is \(-10x\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a number by a sum or a product within parentheses. It states that when you have expressions in the form of \( a(b+c) \), you can distribute \( a \) to both \( b \) and \( c \), resulting in \( ab + ac \). This property is extremely useful in simplifying expressions because it lets you break down complex problems into simpler parts.

In the case of the given exercise, we are applying the distributive property to the expression \(-5(2x)\). The number \(-5\) is distributed to the \(2x\) inside the parentheses. By doing this, we multiply \(-5\) with \(2x\), leading to \(-10x\). This step is vital as it not only simplifies the expression but also opens up the path for solving more complicated problems that involve multiple terms.

Negative Numbers

Negative numbers are numbers that are less than zero. In algebra, they play a significant role, especially when dealing with expressions that require simplification. When multiplying or dividing, it's important to remember the rules associated with negative numbers.

One key rule is that multiplying two negative numbers results in a positive product. However, multiplying a negative number by a positive number yields a negative product. In our example, \(-5(2x)\), the \(-5\) is negative, and so is the product when it multiplies \(2x\), resulting in \(-10x\).

Understanding how negative numbers interact in operations like addition, subtraction, multiplication, and division is essential for solving algebraic expressions effectively. Avoiding common pitfalls with negative signs can help prevent mistakes in both simple and complex calculations.

Algebraic Multiplication

Algebraic multiplication involves multiplying algebraic expressions or terms. This can include terms with variables, constants, and negative numbers. The main goal is to simplify these expressions while keeping track of signs and coefficients.

When multiplying algebraic terms, follow these steps:

• Multiply the coefficients (numbers in front of variables) together.
• Apply the multiplication rules for negative and positive numbers correctly.
• Combine like terms if possible.

In our example, \(-5(2x)\), we multiply the coefficient \(-5\) by \(2\) from \(2x\). The variable \(x\) remains unchanged. This gives us \(-10x\) as the simplified expression.

Mastering multiplication of algebraic expressions is a crucial skill that will help immensely as you progress in algebra. It lays the groundwork for solving equations, understanding functions, and working with polynomials.