Problem 34 - Introductory and Intermediate Algebra for College Students 4th

Question

Use a form of the distributive property to rewrite each algebraic expression without parentheses. $$7(x+y)$$

Step-by-Step Solution

Answer

The expression \(7(x+y)\) when distributed using the property, equals to \(7x + 7y\).

Step 1: Identify the Components of the Algebraic Expression

In this problem, the algebraic expression is \(7(x+y)\). The components to identify are 7, x and y.

Step 2: Apply Distributive Property

The distributive property says you can remove the parentheses by multiplying each part inside the parentheses by the factor outside the parentheses. Do this by multiplying 7 by x and 7 by y.

Step 3: Write the Expression Without Parentheses

After distributing the 7 to the x and y, write the expression without parentheses. This will give you \(7*x + 7*y\), which simplifies to \(7x + 7y\)

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as add, subtract, multiply, and divide). The key to understanding algebraic expressions is recognizing their components and how they're combined. For instance, in the expression 7(x+y), 7 is a numerical factor, while x and y are variables, which can represent various numbers. The parentheses indicate that x and y are being added together before being multiplied by 7.

By learning about algebraic expressions, students lay the foundation for solving equations and developing more advanced math skills. It's important to remember that expressions are not the same as equations; an equation states that two expressions are equal, whereas an expression is simply a value that can be calculated or simplified.

Simplifying Expressions

The process of simplifying expressions is a fundamental skill in algebra which involves reducing an expression to its most basic form. This often makes the expression easier to understand and work with. For simplification, you may combine like terms, use the distributive property, and perform arithmetic operations.For example, using the distributive property in the given exercise 7(x+y) simplifies the expression by removing the parentheses and combining like terms, resulting in 7x + 7y.

To further simplify expressions, always look for opportunities to combine like terms, which are terms that have the same variable raised to the same power. The goal is always to make the expression as straightforward as possible, opening the door to easier manipulation, whether for graphing, solving, or applying to real-world problems.

Intermediate Algebra

The segment of math known as intermediate algebra serves as a bridge between basic algebra and advanced mathematical study. It encompasses a greater depth of algebraic concepts, such as radical expressions, rational expressions, complex numbers, and quadratic equations. The expression 7(x+y) falls into the realm of intermediate algebra due to the presence of variables and the need to apply the distributive property—a technique that is central to this level of algebra.

Intermediate algebra challenges students to deepen their understanding of variable manipulation and to explore the relationships between algebraic expressions. Mastering this subject is essential for success in higher levels of math, including calculus and beyond. It also forms the backbone of many applied sciences, economics, and engineering courses, highlighting the importance of algebra in a broad array of academic and professional fields.