Problem 30 - Algebra 2

Question

Name the property illustrated by each statement. If \(y-2=-8,\) then \(3(y-2)=3(8)\)

Step-by-Step Solution

Answer

Multiplication Property of Equality.

Step 1: Identify the Operation Change

In the given statement, initially, we have the equation \( y - 2 = -8 \). In the next statement, both sides are multiplied by 3, which results in \( 3(y-2) = 3(-8) \). This change involves multiplying both sides of the equation by the same number.

Step 2: Name the Property

The property that allows us to multiply both sides of an equation by the same non-zero number without changing the equality is called the Multiplication Property of Equality.

Key Concepts

Properties of Equality

The properties of equality are fundamental in understanding how to solve and manipulate equations. These properties allow us to perform operations on both sides of an equation while keeping the equation balanced. Imagine an equation as a balanced scale; any operation performed on one side must be counterbalanced by the same operation on the other.
Here are some key properties of equality:

• Reflexive Property: Any quantity is equal to itself. This may seem obvious but is crucial for understanding the concept of equality. For any number\( a \), we have \( a = a \).
• Symmetric Property: This property states that if one quantity equals another, the reverse is also true. For example, if \( a = b \), then \( b = a \).
• Transitive Property: If one quantity equals a second and that second equals a third, then the first equals the third. Mathematically, if \( a = b \) and \( b = c \), then \( a = c \).
• Additive Property of Equality: Adding the same number to both sides of the equation maintains equality.

Understanding these properties is essential for working with equations effectively.

Solving Equations

At the heart of algebra is the practice of solving equations to find the unknown variable, often represented as \( x \) or \( y \). The essence of solving equations involves using properties of equality to isolate the variable so you can determine its value.

• Identify the variable: Determine which symbol in your equation represents the unknown you need to solve for.
• Use inverse operations: Apply operations that "undo" the existing operations around the variable. For instance, if a variable is being added to a number, subtract that number. If it's being multiplied, divide by that number.

By applying these principles step-by-step, you can simplify an equation to the form \( x = a \), making it straightforward to see the solution. For example, to solve the equation \( y - 2 = -8 \), you would add 2 to both sides to get \( y = -6 \). This kind of systematic approach is crucial for solving even the more complex equations.

Equation Manipulation

Equation manipulation refers to the techniques and steps taken to simplify and solve equations. It's a logical procedure ensuring each step taken maintains the equality of the equation using properties of equality.
Key actions in equation manipulation include:

• Combining like terms: Simplify the equation by adding or subtracting similar terms on each side.
• Applying the Multiplication or Division Property of Equality: As demonstrated in the exercise, these properties allow you to multiply or divide both sides of the equation by the same non-zero number.
• Clearing fractions: If an equation includes fractions, multiply through by the least common denominator to eliminate them.

In the given example, applying the Multiplication Property of Equality by multiplying both sides by 3 does not change the equation's truth, but it presents a transformed, yet equivalent equation, \( 3(y-2) = 3(-8) \). This manipulation is key to advancing through numerical steps towards finding a solution.