Problem 37 - Algebra 1: Concepts and Skills

Question

Name the property shown by the statement. $$-3+(-5)=-5+(-3)$$

Step-by-Step Solution

Answer

The property shown by the statement \(-3+(-5)=-5+(-3)\) is the Commutative Property of Addition.

Step 1: Identify the Variables

In the given statement, \(-3+(-5)=-5+(-3)\), we have two terms: -3 and -5. The order of these terms are switched on the two sides of the equation but the overall value of each side remains the same.

Step 2: Identify the Mathematical Property

The mathematical property that allows us to change the order of the terms without changing the sum is the Commutative Property. The commutative property states that changing the order of the operands does not change the result of the operation.

Step 3: Confirm the Property

Putting Step 1 and Step 2 together, we can confirm that the statement, \(-3+(-5)=-5+(-3)\), shows the Commutative Property of Addition.

Key Concepts

Mathematical Properties

Understanding mathematical properties is fundamental for anyone studying arithmetic and algebra. These properties are basic rules that apply to numbers and operations. They make it easier to simplify and solve equations and expressions. Some key properties you'll encounter include the Commutative, Associative, Distributive, and Identity properties. Each has a unique role when it comes to manipulating numbers and can often be applied to ensure calculations are efficient and effective.

For instance, if you understand these properties well, you can easily recognize patterns in numbers and rapidly resolve complex arithmetic problems. This knowledge underpins much of algebra and beyond, as the properties hold true regardless of the complexity of the expressions involved.

Commutative Property

The Commutative Property is one of the cornerstone concepts in basic algebra that deals with the order of operations. Specifically, it refers to the ability to switch or 'commute' numbers around in an addition or multiplication equation without affecting the result.

In the case of addition, the property can be expressed as: \( a + b = b + a \). Similarly, for multiplication, \( a \times b = b \times a \). Understanding this property helps students rearrange the numbers in an equation for convenience while ensuring the integrity of the equation is maintained.

Basic Algebra

At the heart of basic algebra is the manipulation of symbols and letters representing numbers to solve for unknowns. Algebra builds on the properties of arithmetic to deal with more abstract concepts such as variables and expressions. It’s vital to grasp the basic principles, like the Commutative Property, to simplify and solve algebraic problems.

For example, knowing the commutative property can help when rearranging terms in an equation to isolate a variable. Using these foundational rules, students can later tackle more advanced topics in algebra with confidence.

Equality of Expressions

Equality of expressions is an essential concept in algebra which signifies that two expressions represent the same value. An equation, like \( -3 + (-5) = -5 + (-3) \), is an assertion that two different expressions are equivalent. Recognizing the equality of seemingly different expressions often requires applying mathematical properties to manipulate and compare them.

This holds significant implications for solving equations where the goal is often to transform the equation, step by step, maintaining equality until the solution is isolated.