Problem 74 - Algebra 1: Concepts and Skills

Question

Which expression is equivalent to \((x+7) 3 ?\) $$(A) x+21$$ $$(B) 3 x+7$$ $$(C) 3 x+10$$ $$(D) 3 x+21$$

Step-by-Step Solution

Answer

The equivalent expression to \((x+7) 3\) is (D) \(3x + 21\).

Step 1: Apply the distributive property

Multiply each term inside the brackets by 3: \[3*x + 3*7\]

Step 2: Simplify

Perform the multiplication: \[3x + 21.\]

Key Concepts

Equivalent Expressions

In algebra, finding equivalent expressions is a core skill that helps determine different forms of an equation that equal the same value. Equivalent expressions are fundamentally expressions that, despite their different appearances, result in the same value for any variable substitution. For example, if you have

• Expression 1: \(3(x + 7)\)
• Expression 2: \(3x + 21\)

both expressions will yield the same result when the same value is substituted for \(x\).

This is due to the distributive property, which allows you to distribute a term across terms within parentheses. Evaluating each potential expression for equivalence is a crucial part of simplifying problems and is often tested in exercises to ensure understanding of the concept. When identifying equivalent expressions, apply operations correctly and check that simplifications or expansions lead to the same solutions.

Algebraic Expressions

Algebraic expressions are mathematical phrases capturing relationships and operations performed on variables and constants. These expressions are comprised of numbers, variables (like \(x, y\)), and arithmetic operations. For instance, in the expression \(3(x + 7)\), \(x\) is the variable, and 3 and 7 are constants.

They provide a means to represent practical problem scenarios in a generalized form. Understanding how to manipulate and simplify these expressions is vital. It's important to

• Recognize terms: individual parts of an expression separated by plus or minus signs
• Identify coefficients: the numerical part of terms involving variables (e.g., 3 is the coefficient in \(3x\))
• Apply properties to solve or simplify – distributive, associative, and commutative

By mastering these manipulations, students can solve equations and understand dynamic relationships between variables, which is a critical element in algebra.

Basic Algebra Concepts

When delving into algebra, some foundational ideas provide the building blocks for more advanced topics. These include various properties and concepts designed to organize and simplify equations. At the heart of understanding algebra is mastering the use of these properties:

• **The Distributive Property:** Allows you to multiply a number by each term within parentheses, e.g., \(a(b + c) = ab + ac\)
• **Variables and Constants:** Variables are symbols that take on different values, while constants are fixed numbers.
• **Combining like terms:** In an expression, terms that have the same variables raised to the same powers can be combined to simplify the expression.

By breaking down complex problems into these basic components, you can create and manipulate equations more effectively. These concepts serve as essential tools to strategize problem-solving in mathematical scenarios, making what's seemingly complex into manageable and logical steps.
Once mastered, these concepts open the door to tackling a wide range of problems, from simple equations to intricate functions.