Problem 93 - Introductory Algebra for College Students

Question

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 expression is \(2x - 78\).

Step 1: Distribute

We start by distributing the 10 inside the parentheses to each term: \(10 \cdot \frac{x}{5} - 10 \cdot \frac{39}{5}\). This means multiplying each term in the parentheses by 10.

Step 2: Multiply

Perform the multiplication: \(2x - 78\).

Step 3: Simplify

The expression is already in its simplest form, so no further simplification is required.

Key Concepts

Distributive Property

Understanding the distributive property is essential to working with algebraic expressions. It states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. In other words, for any numbers a, b, and c, the property is represented as follows:

\[a(b + c) = ab + ac\].

This property is particularly useful when dealing with expressions inside parentheses. In the exercise \(10\left(\frac{x}{5}-\frac{39}{5}\right)\), we apply the distributive property by multiplying 10 with each term inside the parentheses. This breaks down the problem into simpler parts, making it easier to solve step by step and avoids potential errors with complex fraction operations.

Simplifying Expressions

Simplifying algebraic expressions is a common task which often involves combining like terms and reducing fractions to their simplest form. To combine like terms, we add or subtract the coefficients of the terms that have the exact same variables raised to the same power. Remember that only like terms can be combined; terms with different variables or exponents cannot be simplified in this way.

When simplifying the distributed expression

\(10 \cdot \frac{x}{5} - 10 \cdot \frac{39}{5}\),

we first multiply the numerators by 10, resulting in \(2x - 78\), since \(\frac{10}{5} = 2\). No like terms are left to combine after this step, indicating that the expression is in its simplest form. Simplifying not only makes the equation more manageable but also prepares it for further operations such as solving for variables or evaluating the expression.

Multiplication of Fractions

Multiplication of fractions might seem daunting at first, but it follows a straightforward rule: multiply the numerators together and the denominators together. The general form of this rule is:

\[\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}\].

However, when multiplying a fraction by a whole number, you simply multiply the numerator by the whole number, as the denominator will multiply by 1 (since any whole number can be written as itself over 1). In our exercise, multiplying the whole number 10 by each fraction \(\frac{x}{5}\) and \(\frac{39}{5}\) simplifies down to multiplying the numerators by 10, resulting in 10x over 5 and 390 over 5, which further simplifies to \(2x\) and \(78\), respectively. Here, acknowledging these patterns helps students avoid mistakes and feel more confident in tackling similar problems.