Problem 23 - Algebra 2

Question

Name the property illustrated by each equation. $$ -6 x y+0=-6 x y $$

Step-by-Step Solution

Answer

Additive Identity Property

Step 1: Identify the Equation Form

The given equation is \(-6xy + 0 = -6xy\). Notice that a zero is involved, added to the expression on the left-hand side.

Step 2: Recall Properties of Addition

Remember that in mathematics, adding zero to any number does not change the value of that number. This is called the Additive Identity Property.

Step 3: Recognize the Property Illustrated

The equation \(-6xy + 0 = -6xy\) is an example of the Additive Identity Property because adding zero to \(-6xy\) leaves it unchanged.

Key Concepts

zero in addition

When it comes to addition, zero plays a very special role. This is because zero is the number that doesn't affect other numbers when it’s added. When you add zero to any number, the number stays exactly the same.

For example, if you take the number 5 and add zero, you still have 5. In mathematical terms, this can be written as:

• \(5 + 0 = 5\)

Similarly, if you add zero to a complex expression like \(-6xy\), the expression remains unchanged:

• \(-6xy + 0 = -6xy\)

This behavior of zero is consistent and is why it is known as the "additive identity." This is an important aspect of zero that makes it unique and essential in math.

properties of addition

Addition is not just about combining numbers, but also about understanding certain unique properties that make calculations easier and more predictable.

Some of the key properties of addition include:

• Commutative Property: The order in which two numbers are added does not change the sum. For example, \(a + b = b + a\).
• Associative Property: When you add three or more numbers, the sum is the same no matter how the numbers are grouped. For instance, \((a + b) + c = a + (b + c)\).
• Additive Identity Property: Adding zero to any number gives you the original number. This is exactly what we see in the given equation, \(-6xy + 0 = -6xy\), where zero, our additive identity, leaves the expression unchanged.

These properties are fundamental in understanding and solving mathematical problems, ensuring that no matter how numbers are added, the results are consistent.

mathematical properties

Mathematical properties provide a framework that allows us to solve and manipulate numbers logically and predictably. They are the rules that are always true and apply no matter what numbers we are dealing with.

Key mathematical properties include:

• Additive Identity: As explained before, this is the property where adding zero to any number leaves the number unchanged.
• Multiplicative Identity: This is the property where multiplying any number by one gives the same number. For example, \(a \times 1 = a\).
• Distributive Property: It applies when you multiply a sum by a number, giving the same result as multiplying each addend individually and then adding the products. Expressing this as \(a(b + c) = ab + ac\).

Understanding these mathematical properties is crucial for recognizing patterns and simplifying expressions in algebra and beyond. They not only make complex problems manageable but also reveal the deep interconnectedness present in all areas of mathematics.