Problem 29 - Elementary Algebra

Question

Use the commutative property of addition and multiplication to write expressions for an equal number for the following problems. You need not perform any calculations. $$6(s+1)$$

Step-by-Step Solution

Answer

Question: Rewrite the expression 6(s+1) using the commutative property of addition and multiplication. Answer: 6 + (s×6)

Step 1: Apply the distributive property

Apply the distributive property to the expression $$6(s+1)$$, which means we multiply 6 with both s and 1: $$6(s+1) = 6s + 6$$

Step 2: Apply the commutative property of multiplication

Use the commutative property of multiplication to swap the order of the multiplication between 6 and s: $$6s = s\times6$$ Now, substitute this back into the previous expression: $$6s+6 = (s\times6) + 6$$

Step 3: Apply the commutative property of addition

Use the commutative property of addition to swap the terms in the addition: $$(s\times6) + 6 = 6 + (s\times6)$$ The rewritten expression using the commutative property of addition and multiplication is: $$6(s+1) = 6 + (s\times6)$$

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that connects multiplication and addition. It helps us simplify expressions and solve equations more easily. When we say we are using the distributive property, we mean that we are distributing a single term across terms inside parentheses via multiplication.
For example, in the expression \(6(s+1)\), the number 6 is distributed to the terms inside the parentheses, which are \(s\) and 1. This means we perform the multiplication:

• 6 multiplied by \(s\), resulting in \(6s\)
• 6 multiplied by 1, resulting in 6

Therefore, the expression becomes \(6s + 6\). This illustrates how the distributive property effectively "spreads out" the multiplication over addition within parentheses.

Addition

Addition is one of the basic operations in mathematics, which involves combining numbers or expressions to get a sum. In algebra, addition maintains specific properties that help in simplifying expressions. One of these properties is the commutative property, which states that the order of addition does not affect the sum.
This means that no matter how you add two numbers or terms, the result will be the same. For instance, if you have the expression \(a + b\), it will yield the same value as \(b + a\). Applying this property, you can rearrange terms within an expression to make further calculations clearer.
This can be particularly useful when organizing complex expressions to make them easier to understand and compute.

Multiplication

Multiplication frequently appears in algebra when dealing with variables and numbers. It is an operation where one number, the multiplicand, is increased by itself a specified number of times, indicated by the multiplier. Similar to addition, multiplication also follows the commutative property.
This property allows us to swap the multiplicative terms without affecting the product. For example, \(a \times b\) is equal to \(b \times a\). In the context of algebraic expressions, this freedom to reorder factors can simplify calculations and make expressions more conveniently arranged.
In cases where you express equations or expand expressions, understanding how to apply multiplication and its properties like the commutative property can be very beneficial.

Expressions

In mathematics, an expression is a combination of numbers, variables, and operations like addition or multiplication. Expressions are the building blocks of equations and functions. They help represent mathematical ideas and relationships.
Expressions can be as simple as a single number or more complex with several terms combined. When working with expressions, properties like the distributive, commutative, and associative properties are vital tools for manipulation and simplification.

• For example, using these properties effectively allows us to rewrite complex expressions in simpler forms.
• Understanding expressions and their manipulation is essential in solving equations and understanding mathematical concepts deeply.

Learning to recognize and work with expressions is a foundational skill in mastering algebra and mathematics as a whole.