Problem 68 - College Algebra with Corequisite Support

Question

What property of real numbers would simplify the following expression: \(4+7(x-1) ?\)

Step-by-Step Solution

Answer

The distributive property simplifies the expression to \(7x - 3\).

Step 1: Identify Properties of Real Numbers

Start by considering which properties of real numbers could be used to simplify the expression. Key properties to remember include the commutative, associative, distributive properties, and the identity elements of addition and multiplication.

Step 2: Recognize the Structure of the Expression

The expression is given as \(4 + 7(x - 1)\). Notice that it involves a constant (4) and a term \(7(x-1)\), where 7 is being multiplied by the expression \((x-1)\). This structure suggests the use of the distributive property, which is useful when dealing with multiplication across addition or subtraction.

Step 3: Apply the Distributive Property

Apply the distributive property to the expression \(7(x-1)\). According to the distributive property, \( a(b + c) = ab + ac \). For our expression, distribute the 7 across the \(x-1\): \(7(x - 1)\) becomes \(7x - 7\).

Step 4: Rewrite the Expression

Substitute the expression \(7(x-1)\) by its distributed equivalent \(7x - 7\) back into the original problem: \(4 + 7(x-1)\) becomes \(4 + 7x - 7\).

Step 5: Simplify the Expression

Combine the constant terms in the expression \(4 - 7\) to further simplify it. This results in \(7x - 3\). This simplification highlights that the distributive property was key to transforming and simplifying the expression efficiently.

Key Concepts

Distributive Property

The distributive property is a fundamental component of real number operations that presents itself frequently in algebra. To put it simply, the distributive property allows us to multiply a single term by the terms inside a parenthesis, facilitating easier manipulation of expressions. This property is especially useful when dealing with multiplication across sums or differences.

In mathematical terms, the distributive property states that for any three numbers, say, \(a\), \(b\), and \(c\), the expression \(a(b+c)\) can be expanded to \(ab+ac\). In our exercise, we're prompted to apply it to the expression \(7(x-1)\). Here, 7 is distributed across \(x\) and \(-1\).

• Begin by multiplying 7 by \(x\) to obtain \(7x\).
• Next, multiply 7 by \(-1\) to achieve \(-7\).

This results in the expression \(7x - 7\), clearly showcasing how the distributive property helps break complex expressions into simpler parts, conducive to further algebraic operations.

Algebraic Simplification

Algebraic simplification plays a critical role in making mathematical expressions simpler and more manageable. It often involves combining like terms, reducing expressions, and rearranging them to their most efficient forms. In the context of our problem, after applying the distributive property, we arrive at the expression \(4 + 7x - 7\).

To simplify this further, we identify and combine the constant terms: 4 and \(-7\).

• These constants, also called like terms, can be directly added or subtracted.
• Calculating \(4 - 7\) results in \(-3\).

Thus, the expression \(4 + 7x - 7\) becomes \(7x - 3\), presenting it in its simplified form. This process of algebraic simplification can significantly aid in solving equations and making calculations more straightforward.

Expression Structure

Understanding the structure of an expression is vital in determining how to approach its simplification. An expression's structure can reveal which mathematical properties and operations are most applicable. For our given expression, \(4 + 7(x-1)\), a quick analysis shows the existence of two major parts: a constant 4, and the expression \(7(x-1)\).

The presence of a term that involves multiplication across a subtraction hints at the necessity of the distributive property. Recognizing this structure allows us to strategically apply appropriate algebraic properties, such as the distributive property in this case, to transform the expression into simpler components.

• This recognition is important as it paves the way for effective algebraic simplification.
• Careful observation can prevent missteps and lead to more efficient problem-solving techniques.

By analyzing expression structures, students can discern the connections between various parts of an expression and the properties that govern them, laying the groundwork for more advanced algebraic skills.