Problem 40 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$ 5\left(\frac{1}{10} x-\frac{2}{15}\right) $$

Step-by-Step Solution

Answer

The simplified expression without parentheses is \(\frac{1}{2}x - \frac{2}{3}\).

Step 1: Distribute the 5 to each term inside the parentheses

Here, \(a = 5, b = \frac{1}{10}x, c = -\frac{2}{15}\). According to the distributive property, \(5\left(\frac{1}{10}x-\frac{2}{15}\right) = 5 * \frac{1}{10}x - 5*\frac{2}{15}\)

Step 2: Multiply 5 with each term

The multiplication results in \(\frac{1}{2}x - \frac{2}{3}\)

Step 3: Final Simplified Expression

The final simplified expression without parentheses is \(\frac{1}{2}x - \frac{2}{3}\)

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions is an essential skill in algebra. It involves reducing an expression to its most basic form while maintaining its value. The process eliminates complexity, making the expression easier to work with, whether you're evaluating, solving, or graphing it.

Imagine you have a messy bedroom; simplifying expressions is like cleaning it up so that you can move around easily without stepping on anything. You can simplify expressions by combining like terms, using the distributive property to remove parentheses, and reducing fractions. For instance, in our example, the expression was simplified by distributing the 5 and then combining the resulting terms.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. Variables represent unknown values and are often shown as letters like 'x' or 'y'. Numbers are the constants, and operations include addition, subtraction, multiplication, and division.

An example of an algebraic expression is \( 5\left(\frac{1}{10} x-\frac{2}{15}\right) \), where \( x \) is the variable. These expressions are the backbone of algebra and are used to describe relationships between quantities. When you understand how to work with them, you're better equipped to tackle equations and real-world problems.

Properties of Operations

The properties of operations are rules that help us manipulate numbers and variables efficiently. These properties include the associative, commutative, distributive, and identity properties. Each property has its own 'superpower' in making calculations easier.

For instance, the distributive property allows you to multiply a single term across terms within parentheses, like a magic pass that lets you distribute something evenly to everyone in a group. In our exercise, we used this property to distribute the 5 across each term inside the parentheses, simplifying the expression to something more manageable.

Basic Algebra

Basic algebra is the foundation upon which more complex mathematical concepts are built. It deals with solving for unknowns, working with expressions, and understanding the rules that govern these manipulations.

Knowing how to simplify expressions, handle algebraic expressions, and apply properties of operations primes you for success in algebra. It's like learning the grammar of mathematics; once you have a strong grasp of the basics, you can start 'speaking' more complex 'sentences' or solving more intricate problems. Despite being 'basic,' this knowledge is powerful and versatile, useful in a wide range of applications, from everyday problem-solving to advanced science and engineering tasks.