Problem 70 - Algebra 2

Question

Name the property illustrated by each statement. If \(3 x=4 y\) and \(4 y=15 z,\) then \(3 x=15 z\)

Step-by-Step Solution

Answer

Transitive Property of Equality.

Step 1: Identify the Given Equations

The problem statement provides us with two separate equations: \( 3x = 4y \) and \( 4y = 15z \). Our goal is to examine how they relate to the resulting equation \( 3x = 15z \).

Step 2: Analyze the Transition Between Equations

Notice that from the initial equations \( 3x = 4y \) and \( 4y = 15z \), we are essentially linking them to get \( 3x = 15z \). This is done by equating the expressions for \( 4y \) in both equations. Since both are equal to \( 4y \), you can substitute directly.

Step 3: Name the Property Used

This method of deriving \( 3x = 15z \) from \( 3x = 4y \) and \( 4y = 15z \) uses the Transitive Property of Equality. This property states that if \( a = b \) and \( b = c \), then \( a = c \).

Key Concepts

Algebraic Properties

Algebraic properties are foundational concepts in mathematics that allow us to perform operations on equations with confidence and consistency. One of the essential properties is the Transitive Property of Equality, which is used extensively in algebra. This property states that if two things are equal to a third thing, then the first two things are equal to each other as well. For example, if we know that \( a = b \) and \( b = c \), then it naturally follows that \( a = c \). This logical consistency ensures that mathematical reasoning is reliable and predictable.

Some other common algebraic properties include:

• Reflexive Property: This states that any value is equal to itself, expressed as \( a = a \).
• Symmetric Property: If \( a = b \), then it also holds that \( b = a \).
• Distributive Property: This involves operations over addition, such as \( a(b + c) = ab + ac \).

These properties form the backbone of solving equations and understanding algebraic expressions, providing a toolkit that can be applied to various mathematical problems.

Equations

Equations are mathematical statements asserting equality between two expressions. They are a fundamental part of algebra, allowing us to represent relationships between quantities and solve for unknown values. An equation consists of two expressions separated by an equal sign, such as \( 3x = 4y \) or \( 4y = 15z \).

There are several types of equations:

• Linear Equations, which involve expressions of the first degree (e.g., \( ax + b = c \)).
• Quadratic Equations, which include squares of variables (e.g., \( ax^2 + bx + c = 0 \)).
• Polynomial Equations, which are higher degree equations (e.g., \( ax^n + bx^{n-1} + \, ... \, + d = 0 \)).

Solving equations often involves finding the value(s) of unknown variables that make the equation true. Techniques like substitution, elimination, and graphing are often employed to find solutions. Equations allow us to explore and understand complex mathematical concepts and real-world problems.

Substitution Method

The substitution method is a technique used to solve systems of equations, especially when dealing with two or more equations simultaneously. It involves replacing one variable with an equivalent expression to simplify the system and solve for the unknowns. This method is especially handy when equations are already set up for substitution.

Here's how you can approach it:

• Start by solving one of the equations for one variable in terms of the other. For instance, if you have the equation \( x = 2y + 3 \), you express \( x \) in terms of \( y \).
• Substitute the expression found into the other equation. Using \( x = 2y + 3 \), if the other equation is \( x + y = 7 \), you would substitute \( 2y + 3 \) for \( x \).
• Solve the resulting single-variable equation. After substitution, you have a simpler equation that you can solve for one variable.
• Finally, plug the solved value back into one of the original equations to find the other missing value.

The substitution method is powerful because it breaks down complex systems into manageable parts, making the solving process more straightforward.