Problem 46 - Intermediate Algebra

Question

Replace each question mark to make the given statement an application of the indicated property of equality. For example, \(16=\) ? becomes \(16=16\) because of the reflexive property of equality. If \(y=x\) and \(x=z+2\), then \(y=?\) (Transitive property of equality)

Step-by-Step Solution

Answer

y = z + 2.

Step 1: Understand the Transitive Property of Equality

The Transitive Property of Equality states that if two values are both equal to a third value, then they are equal to each other. In mathematical terms, if \( a = b \) and \( b = c \), then \( a = c \).

Step 2: Identify Given Equations

In the problem, we are given that \( y = x \) and \( x = z + 2 \). We need to find an expression for \( y \) using these equations.

Step 3: Apply the Transitive Property

According to the transitive property, since \( y = x \) and \( x = z + 2 \), we can conclude that \( y = z + 2 \). By substituting \( x \) in the equation \( y = x \) with \( z + 2 \), the equality is maintained.

Key Concepts

Understanding Algebraic Properties

Algebraic properties are essential rules that form the foundation of algebra. They allow us to manipulate equations and expressions to find solutions. One of the key properties is the Transitive Property of Equality.

• Reflexive Property: This property states that any number is equal to itself. For example, 5 = 5.
• Symmetric Property: If a = b, then b = a. This means equations can be flipped or reversed.
• Transitive Property: If a = b and b = c, then a = c. It creates a link between equal expressions or values.

By understanding these properties, solving algebraic equations becomes a systematic process, allowing us to find unknown values with confidence.

Approaching Problem Solving

Solving mathematical problems often requires a strategic approach where you use known information to derive new insights. In the exercise given, the task is to apply the Transitive Property of Equality to find the value of \(y\). To effectively solve equations:

• Identify what you know: Write down the equations provided or break down any text into mathematical statements.
• Look for connections: Determine how different equations relate to each other. For instance, find a common variable.
• Apply logical reasoning: Use algebraic properties to transform and simplify equations.

This methodical problem-solving technique will serve any student well in both algebra and other areas that require critical thinking.

Working with Equations

Equations are mathematical statements that show the equality between two expressions. To solve for any variable, one must manipulate these equations while keeping them balanced.In our given problem:- You start with two equations: \( y = x \) and \( x = z + 2 \).- By applying the Transitive Property of Equality, you substitute \( x \) in \( y = x \) with \( z + 2 \), which results in the equation: \( y = z + 2 \).This demonstrates how solving equations involves substituting known values or expressions to find unknown variables. Learning to work effectively with equations empowers students to tackle a variety of mathematical challenges.