Problem 81 - Beginning and Intermediate Algebra

Question

Name the properties illustrated by each true statement. See Example 6 \(-4(y+7)=-4 \cdot y+(-4) \cdot 7\)

Step-by-Step Solution

Answer

The statement illustrates the Distributive Property of Multiplication over Addition.

Step 1: Identify the operation used

Look at the expression on the left-hand side of the equation: \[-4(y+7)\]The left side uses the distributive operation where a single number is multiplying an entire sum inside the parentheses.

Step 2: Identify the transformation on the right-hand side

Observe the right-hand side of the equation: \[-4 \cdot y + (-4) \cdot 7\]Notice that the operation distributes the multiplication of \(-4\) over each term inside the parentheses \((y + 7)\).

Step 3: Deduce the property illustrated

The equation shows a transformation where multiplication is distributed over addition. This is known as the Distributive Property of Multiplication over Addition. It states that multiplying a number by a sum is the same as multiplying each addend individually and then adding the products.

Key Concepts

Algebraic Properties

Algebraic properties are fundamental rules in algebra that help in simplifying expressions and solving equations. One of the key algebraic properties is the Distributive Property. This property allows us to eliminate parentheses in expressions by distributing a multiplication operation across the terms inside the parentheses. For example, in the expression \[-4(y+7)\], the number \(-4\)is distributed to both \(y\)and \(7\), resulting in \(-4 \cdot y + (-4) \cdot 7\).

• Commutative Property: Changes the order of operands, applicable for addition and multiplication, e.g., \(a + b = b + a\).
• Associative Property: Changes the grouping of numbers, also for addition and multiplication, e.g., \((a + b) + c = a + (b + c)\).
• Identity Property: Deals with the identity element, e.g., \(a \cdot 1 = a\).

Understanding these properties is crucial as they enable the simplification and solving of equations efficiently.

Equations

Equations are mathematical statements that assert the equality between two expressions. They play a crucial role in algebra, allowing us to find unknown values. An equation typically consists of two expressions separated by an equals sign \(=\). The goal is often to solve for a variable, which represents an unknown number. For example, in \(-4(y+7)=-4 \cdot y + (-4) \cdot 7\), both expressions are equal, demonstrating the Distributive Property.

• Linear Equations: These have variables raised to the power of one, having the simplest form \(ax + b = c\).
• Quadratic Equations: Involve squared variables, such as \(ax^2 + bx + c = 0\).
• Systems of Equations: Sets of equations solved together to find a common solution.

To solve equations efficiently, one must be comfortable manipulating both sides using algebraic properties and operations.

Mathematical Operations

Mathematical operations are processes or actions that are carried out on numbers or expressions. These include addition, subtraction, multiplication, and division—each having specific rules and properties. A solid understanding of these operations aids in performing more complex algebraic manipulations and solving equations. For instance, the multiplication operation in expressions like \(-4(y + 7)\) requires distributing \(-4\) to both \(y\) and \(7\), illustrating how understanding operations facilitates applying algebraic properties such as the Distributive Property.

• Addition: Combining numbers to get a sum.
• Subtraction: Finding the difference between numbers.
• Multiplication: Calculating the product, often used in distributing properties.
• Division: Splitting into equal parts or determining how many times one number fits into another.

These operations are the backbone of mathematics, enabling us to explore various concepts and solve a myriad of problems efficiently.