Problem 91 - Introductory Algebra for College Students

Question

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. I did not use the distributive property to simplify \(3(2 x+5 x)\)

Step-by-Step Solution

Answer

The statement 'I did not use the distributive property to simplify \(3(2x+5x)\)' does not make sense because the distributive property was clearly used to simplify the expression to \(21x\).

Step 1: Apply the Distributive Property

Apply the distributive property according to the operation \(a(b + c) = ab + ac\). Using this rule, apply the distributive property to simplify \(3(2x+5x)\). This will result in \(3*2x + 3*5x\).

Step 2: Simplify the Expression

Work out the multiplication in the expression. This will result in \(6x + 15x\).

Step 3: Combine Like Terms

Add the two terms which are of the same kind (like terms). This will result in \(21x\).

Step 4: Determine the Sensibility of the Statement

After simplifying the expression by using the distributive property and combining like terms, \(3(2x+5x)\) was reduced to \(21x\). So it's clear that the distributive property was used to simplify the expression, meaning the statement 'I did not use the distributive property to simplify \(3(2x+5x)\)' does not make sense as it contradicts the procedure applied.

Key Concepts

Combining Like Terms

When working with algebraic expressions, combining like terms is a fundamental technique for simplification. Like terms are terms that contain the same variables raised to the same power. For instance, in the expression \(6x + 15x\), both terms are like terms because they each contain the variable \(x\) to the first power.

The process of combining like terms involves adding or subtracting the coefficients (the numerical factors) of these terms. Here's how you do it:

• Identify like terms in the expression.
• Add or subtract the coefficients of these terms.
• Keep the common variable and exponent unchanged.

Applying this to our example, we combine \(6x\) and \(15x\) by adding their coefficients (6 and 15) to get \(21x\). This simplification is essential for solving equations efficiently and understanding algebraic manipulations.

Simplifying Expressions

The goal of simplifying expressions in algebra is to rewrite them in the simplest form while keeping their values unchanged. Simplifying can involve a range of operations, including combining like terms, using the distributive property, and performing arithmetic operations.

To simplify an expression, follow these general steps:

• Distribute any grouped expressions, eliminating parentheses where possible.
• Combine like terms by adding or subtracting them as demonstrated earlier.
• Perform any other arithmetic operations such as multiplying or dividing.

In our example, the expression \(3(2x+5x)\) was simplified by using the distributive property first, followed by combining like terms, resulting in a much simpler expression, \(21x\). Simplification makes expressions cleaner and easier to work with in subsequent algebraic processes.

Algebraic Properties

Understanding algebraic properties is crucial when manipulating and simplifying algebraic expressions. These properties are rules that allow you to rearrange and combine numbers and variables while maintaining the equivalence of the expressions. Some key properties include the Commutative Property (the order in which you add or multiply numbers doesn't affect the sum or product), the Associative Property (rearranging grouping symbols doesn't affect the sum or product), and the Distributive Property which we applied in our example.

The Distributive Property states that \(a(b+c) = ab + ac\). It is used to multiply a single term across terms within parentheses and is vital for simplifying complex expressions. By applying these properties appropriately, you can effectively solve algebraic expressions and equations. Always ensure to apply the correct property relevant to the operation you are performing during simplification.