Problem 5 - Algebra 1: Concepts and Skills

Question

Match the property with the statement that illustrates it. Property of negative one A. \(-1 \cdot 9=-9\) B. \(4(-2)=(-2) 4\) C. \(0 \cdot 8=0\) D. \(1 \cdot(-15)=-15\) E. \(-7(5 \cdot 2)=(-7 \cdot 5) 2\)

Step-by-Step Solution

Answer

The property of -1 matches with statement A (\(-1 \cdot 9=-9\))

Step 1: Review the options

Look at each of the choices given, and identify if it embodies the property of -1. The property of -1 is such that when any number multiplies with -1, the sign of that very number changes. For example, if it's a positive number, it will become negative and if it's already negative, it will transform into positive.

Step 2: Match the property

By comparing the property of -1 with the given options, one can see that option A (\(-1 \cdot 9=-9\)) perfectly illustrates this property. Here -1 is multiplied with 9 to give -9, thus indicating a change in the sign. Thus the property of -1 is matched to statement A.

Key Concepts

Multiplicative Identity

In mathematics, the **Multiplicative Identity** is a fundamental property that involves the number 1. This property states that any number, when multiplied by 1, will remain unchanged. Imagine multiplying any number by 1 as telling that number, "You stay exactly as you are!"
For example:

• When you multiply 5 by 1, the result is still 5, written as: \(1 \cdot 5 = 5\).
• Similarly, when you multiply -8 by 1, it remains -8: \(1 \cdot (-8) = -8\).

This property is important because it helps to maintain the value of numbers across equations and expressions. Regardless of whether the number is positive, negative, or even a fraction, multiplying by 1 keeps it stable.

Zero Property of Multiplication

The **Zero Property of Multiplication** is another key concept in mathematics. This property tells us that the result of multiplying any number by zero is always zero. It acts like an eraser by wiping away any number or variable’s value it multiplies.
Here’s how it works:

• If you multiply 7 by 0, you get 0: \(7 \cdot 0 = 0\).
• Multiplying -3 by 0 also gives 0: \((-3) \cdot 0 = 0\).

This property holds true for any real number, whether positive, negative, or zero itself. It’s a useful tool, especially when solving algebraic expressions, as it can simplify complicated parts of an equation in an instant.

Commutative Property of Multiplication

The **Commutative Property of Multiplication** focuses on the order of multiplication. This property states that the order in which you multiply numbers doesn’t affect the product, thereby providing flexibility in calculations.
For example:

• Consider multiplying 3 and 4. Whether you calculate \(3 \cdot 4\) or \(4 \cdot 3\), the result is always 12.
• Similarly, \((-2) \cdot 6\) equals the same as \(6 \cdot (-2)\), both resulting in -12.

This property is particularly useful in algebra, allowing you to rearrange terms to simplify problems or make complex calculations more manageable. It’s one of the reasons why multiplication is often easier than it first seems, as you can move numbers around to suit the situation without changing the answer.