Problem 40 - 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. $$(5)(b-6)$$

Step-by-Step Solution

Answer

Question: Using the commutative property of addition and multiplication, rewrite the expression \((5)(b-6)\) as an equal expression. Answer: The equal expression is \(5b - 6 \cdot 5\).

Step 1: Identify the given expression

The given expression is \((5)(b-6)\), where we need to apply the commutative property of addition and multiplication.

Step 2: Distributive property for multiplication

First, apply the distributive property for multiplication to expand the given expression: $$(5)(b-6) = 5b - 5 \cdot 6.$$

Step 3: Apply the commutative property of multiplication

Now, use the commutative property of multiplication to rewrite the expression as: $$5b - 5 \cdot 6 = 5b - 6 \cdot 5.$$

Step 4: Write the final equal expression

Based on the rearrangements made using the commutative property, the new equal expression will be: $$5b - 6 \cdot 5.$$

Key Concepts

Addition

In mathematics, addition is one of the basic operations that combines two or more numbers to get a sum. The commutative property of addition states that the order in which you add numbers doesn't change the sum. For example, if you have two numbers, such as 3 and 5, you can add them in any order.

- 3 + 5 = 8 - 5 + 3 = 8
Both calculations result in the same sum of 8. This property makes addition flexible and helps simplify expressions and calculations. It's important to note that the commutative property only applies to addition and multiplication, not to subtraction or division.

Addition's commutative nature ensures that no matter how you group numbers during the calculation process, you'll end up with the same result, making mental math much easier.

Multiplication

Just like addition, multiplication has a commutative property. This means the order of multiplying two numbers can be switched without affecting the result. So, for example:

• 2 \(\times\) 4 = 8
• 4 \(\times\) 2 = 8

In each case, the product is the same, which illustrates that 2 \(\times\) 4 equals 4 \(\times\) 2.

The commutative property of multiplication is immensely useful, especially in algebra, where rearranging terms can simplify solving problems involving variables.

In our original problem, by applying this concept, the expression 5 \(\times\) 6 can be rewritten as 6 \(\times\) 5 without changing its value. This reassures learners that the sequence of numbers being multiplied does not need to be rigid – swapping them around can often reveal simpler pathways to an answer.

Distributive Property

The distributive property is a very useful tool in algebra which helps in simplifying expressions and solving equations. It allows you to multiply a single term by two or more terms inside a parenthesis. For example:

If you have an expression like \((a)(b + c)\), applying the distributive property means:

• \((a)(b + c) = a \cdot b + a \cdot c\)

In essence, you distribute the multiplication over each term in the parenthesis.

In the original step-by-step solution, the expression \((5)(b-6)\) was expanded using the distributive property as follows:

• \(5(b-6) = 5 \times b - 5 \times 6\)

This split simplifies the expression and can make further manipulations possible by using other properties like the commutative property.

Understanding the distributive property not only helps you break down complex expressions into manageable parts but also allows you to expand and simplify expressions while preserving equality.