Problem 5 - Elementary Algebra

Question

Fill in the \((\quad)\) with the proper number or letter so as to make the statement true. Use the commutative properties. $$4(k-5)=(\quad) 4$$

Step-by-Step Solution

Answer

Question: Fill in the blank using the commutative property of multiplication: $4(k-5) = (k-5)\underline{\hspace{1.5cm}}$ Answer: $4$

Step 1: Identify the commutative property

In this exercise we need to utilize the commutative property of multiplication, which states that for any two numbers \(a\) and \(b\), \(ab = ba\).

Step 2: Apply the commutative property

We have the expression \(4(k-5)\). To use the commutative property of multiplication, we can simply switch the two factors: \((k-5)4\). This means that the expression can also be written as \((k-5)4\) without changing its value.

Step 3: Fill in the blank

Now we can fill in the blank with the proper number or letter to make the statement true: $$4(k-5)=(k-5) 4$$

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. These expressions represent a value in a mathematical formula. Let's break it down: an expression like \(4(k-5)\) includes both a constant (4) and a variable expression \(k-5\).
\Variables like \(k\) can take different numbers as values. This makes algebraic expressions flexible for use in equations and inequalities. They're the building blocks for expressing mathematical ideas.
\Performing operations on algebraic expressions often requires the usage of properties from arithmetic and algebra. These properties help simplify, rearrange, or solve expressions. Being familiar with algebraic expressions is crucial for solving equations like \(4(k-5) = (\quad)4\) with confidence.

Properties of Multiplication

Understanding the properties of multiplication is essential for manipulating algebraic expressions. One important property is the commutative property, which states that the order of multiplying numbers doesn't affect the end result. For example, \(ab = ba\).
\When you're dealing with expressions like \(4(k-5)\), the commutative property allows us to rewrite it as \( (k-5)4 \). This swap doesn't change the evaluation of the expression; it's still the same value.
\This property is particularly useful when solving equations because it offers flexibility in organizing terms in a way that might make simplification or further operations easier.
\Additionally, there are other useful properties such as:

\
• The associative property: allows rearranging the grouping of numbers.
\
• The distributive property: helps in expanding expressions.
\

Becoming comfortable with these properties enriches your understanding of how multiplication functions across different situations.

Mathematical Operations

Mathematical operations involve the basic functions of arithmetic: addition, subtraction, multiplication, and division. These operations form the foundation of more complex mathematics, including the manipulation of algebraic expressions and equations.
\In algebra, operations become more intricate when combined with variables. Evaluating expressions involves applying operations according to standard arithmetic rules, often alongside algebraic properties.
\In our exercise of \(4(k-5) = (\quad)4\), the operation of multiplication is used alongside the commutative property to rearrange terms effectively. Here are some tips for handling operations:

\
• Always follow the order of operations (PEMDAS/BODMAS).
\
• Apply properties of operations carefully to simplify or rearrange expressions.
\
• Understand how operations function individually and together to solve problems.
\

This fundamental understanding of mathematical operations is key to mastering algebra and other higher-level mathematics.