Problem 93 - Introductory and Intermediate Algebra for College Students 4th

Question

Exercises \(91-93\) will help you prepare for the material covered in the next section. Multiply and simplify: \(10\left(\frac{x}{5}-\frac{39}{5}\right)\)

Step-by-Step Solution

Answer

The simplified form of the given expression is \(2x - 78\).

Step 1: Apply Distributive Law

Use the distributive property of multiplication over subtraction. This means multiplying 10 with each term in the bracket i.e. \(\frac{x}{5}\) and \(-\frac{39}{5}\). The expression becomes: \(10 * \frac{x}{5} - 10 * \frac{39}{5}\).

Step 2: Multiply

In this step, perform the multiplication. \(10 * \frac{x}{5}\) becomes \(2x\) and \(10 * \frac{39}{5}\) becomes \(78\). Now the expression simplifies to: \(2x - 78\)

Step 3: Final Solution

The final simplified expression after having performed multiplication and simplification is \(2x - 78\).

Key Concepts

Algebraic Expressions

In algebra, expressions are mathematical phrases involving numbers, variables, and operations. They do not have an equal sign, distinguishing them from equations. In the given exercise, the expression is \(10\left(\frac{x}{5}-\frac{39}{5}\right)\). Here, \(x\) is the variable, and \(\frac{x}{5}\) and \(\frac{39}{5}\) are terms contained within the brackets. The goal is to manipulate and simplify these terms to make them more manageable.

Understanding algebraic expressions is essential as they are the foundation for solving equations and inequalities. When handling expressions, it helps to identify the key components:

• **Variables** - Symbols that stand for unknown numbers. In this example, it's \(x\).
• **Constants** - Numbers on their own. For instance, \(39\) in the expression.
• **Coefficients** - Numbers multiplying the variables, like \(10\), which we use in the distribution.
• **Operators** - Indicate the operations to perform, such as the minus sign which suggests subtraction.

Brackets in an expression group terms together and influence the order of operations, making it vital to handle them precisely before further simplification.

Multiplication Simplification

The simplification process involves reducing expressions to their simplest form. This exercise demonstrates `Multiplication Simplification` using the distributive property. The property lets us multiply a single term across all terms in a bracket. Here, we had \(10\left(\frac{x}{5}-\frac{39}{5}\right)\).

First, we need to distribute \(10\) to each term within the brackets:

• Multiply \(10\) by \(\frac{x}{5}\) to get \(2x\).
• Multiply \(10\) by \(\frac{39}{5}\) to get \(78\).

This step-by-step multiplication helps in transforming the expression into \(2x - 78\). Notice how distributing the \(10\) multiplied with fractions involves division, which simplifies quickly when coupled with multiplication. Understanding this process is crucial for reducing and resolving more complex algebraic expressions effectively.

Pre-Algebra Skills

Solid pre-algebra skills are your stepping stone to grasp more complex math concepts. Exercises like multiplying and simplifying algebraic expressions lay the groundwork for future math challenges. Strong pre-algebra skills involve mastering basic arithmetic operations and understanding properties like the distributive law, seen in this exercise.

Here's how the distributive property pops up in a pre-algebra context:

• The property states multiplying a sum by a number is the same as doing each multiplication separately. That's how we went from \(10\left(\frac{x}{5}-\frac{39}{5}\right)\) to \(10 \cdot \frac{x}{5} - 10 \cdot \frac{39}{5}\).
• Working with fractions becomes simpler by reducing them common terms, shown when \(10\cdot \frac{x}{5}\) becomes \(2x\).

Focusing on these skills helps in building confidence for tackling math in high school and beyond. Each step in this simplification exercise enhances your ability to approach algebraic equations with ease and efficiency, boosting your problem-solving dexterity.