Problem 67 - Prealgebra and Introductory Algebra

Question

Use the given property of addition to complete the statement. The Addition Property of Zero $$-13+?=-13$$

Step-by-Step Solution

Answer

The number that makes the given statement \(-13+?=-13\) true, when using the Addition Property of Zero, is \(0\).

Step 1: Recognize The Addition Property of Zero

First, understand the Addition Property of Zero, which states that adding zero to any number yields the same numbere. In other words, for any real number \(a\), the equation \(a + 0 = a\) holds true.

Step 2: Set Up the Equation

Next, set up the equation that is given in the exercise. The equation is \(-13+?=-13\). To solve this equation, one would need to find the value that replaces the question mark \(?\) which makes the statement true.

Step 3: Apply The Addition Property of Zero

Then, apply the Addition Property of Zero to the equation. The unknown number added to \(-13\) to result in \(-13\) must be zero. Therefore, the question mark (?) should be replaced by \(0\).

Key Concepts

Zero Property of Addition

The Zero Property of Addition is a fundamental concept in basic algebra that is very easy to understand and incredibly useful in mathematical calculations. Simply put, this property tells us that when you add zero to any real number, the result is the number itself. This can be visualized with the formula:

• For a number, say, \( a \), the expression \( a + 0 = a \) always holds true.
• Similarly, \( 0 + a = a \) is valid as well.

Understanding this property is essential because it often appears in equations where simplifying expressions is necessary. It helps us realize that zero is a neutral element in addition, maintaining the identity of real numbers.

Real Numbers

Real numbers include all the numbers we commonly use and encounter in everyday life, encompassing both rational and irrational numbers. They can be represented on a number line, and include:

• Rational numbers, such as fractions (e.g., \( \frac{1}{2} \)) and integers (e.g., \( -3, 0, 5 \)).
• Irrational numbers, such as \( \pi \) or the square root of 2, which cannot be expressed as a fraction.

The Addition Property of Zero applicable to all real numbers, meaning regardless of whether you're dealing with positive or negative numbers, whole numbers or fractions, adding zero will leave the number unchanged. This applies in real-world situations and mathematical problems, providing a consistent foundation across various applications.

Basic Algebra

Basic algebra involves a lot of rules and properties which simplify problem solving. The Zero Property of Addition is one of these vital rules. Recognizing and understanding these properties can make tasks in algebra more straightforward.

• Essential skills include solving equations, often requiring the rearrangement of terms.
• In equations like \[-13 + ? = -13\], knowing this property immediately solves for \( ? = 0 \) as adding zero does not change the value.

Employing basic algebraic properties not only simplifies calculations but also builds a strong base for more advanced math topics.

Mathematical Properties

Mathematical properties are rules that apply to numbers and operations, offering a framework within which arithmetic and algebra calculations unfold. Some key properties include:

• The Commutative Property, allowing the order of numbers to change without affecting the result.
• The Associative Property, permitting the grouping of numbers to vary.
• The Identity Property, which includes the Zero Property of Addition, ensures adding zero leaves the original number unchanged.

Understanding these properties enhances mathematical fluency, making it easier to manipulate and solve equations. They are not merely abstract concepts—they are tools that simplify complex problems, ensuring calculations are accurate and efficient.