Problem 1 - Beginning and Intermediate Algebra

Question

If \(z\) varies directly as \(y,\) then as \(y\) increases, the value of \(z=\)_________

Step-by-Step Solution

Answer

As y increases, the value of z increases as well.

Step 1: Understand Direct Variation Relationship

Direct variation means that one variable is proportional to the other variable. In this case, z is directly proportional to y. Mathematically, this can be written as: \(z = ky\) where k is the constant of proportionality.

Step 2: Analyze the Effect of Increasing y on z

Since z is directly proportional to y, if y increases, the value of z will also increase. To further understand this, we can think of the constant of proportionality k as a multiplier. When y gets larger, the product of k and y (which equals z) will also get larger. Therefore, as y increases, the value of z increases as well.

Key Concepts

Proportional Relationship

In mathematics, a **proportional relationship** is a relationship between two quantities where the ratio between them remains constant. If one variable increases, the other variable will also increase proportionally, and vice versa. This relationship is often described by saying one variable "varies directly" as the other.
For example, if a variable \( z \) is directly proportional to another variable \( y \), we say that \( z \) varies directly as \( y \). This is expressed with the equation \( z = ky \), where \( k \) is the constant of proportionality.

• The equation shows that as \( y \) changes, \( z \) changes in a predictable way.
• This kind of relationship is linear, forming a straight line when graphed with \( z \) on one axis and \( y \) on the other.

Understanding proportional relationships is crucial in solving problems related to direct variations, as it allows us to predict the changes in one variable when the other changes.

Constant of Proportionality

The **constant of proportionality**, often represented by \( k \), is a key component in directly proportional relationships. It serves as the multiplier that connects the two variables, allowing for one variable to change in tandem with the other. This constant remains the same no matter the values of \( y \) and \( z \).

In the equation \( z = ky \):

• \( k \) represents the constant of proportionality.
• It determines how much \( z \) changes when \( y \) changes.
• For example, if \( k = 2 \), it means that \( z \) is always twice the value of \( y \).

The constant of proportionality is fundamental in equations and can vary greatly depending on the specific problem or application. Recognizing this constant helps us understand the specific nature of the proportional relationship at play.

Algebraic Expressions

**Algebraic expressions** are a primary way to represent relationships and changes between variables in mathematics. In the context of direct variation, an algebraic expression allows us to articulate the relationship between two variables clearly.
For instance, if \( z \) varies directly with \( y \), the algebraic expression will be \( z = ky \). This expression concisely represents the relationship between \( z \) and \( y \), showing the direct variation influenced by the constant \( k \).

• Algebraic expressions can involve numbers, variables, and arithmetic operations.
• They serve as a mathematical means to describe how quantities change relative to each other.
• These expressions are adaptable and can be used in various situations to describe a wide range of proportional relationships.

Learning to work with algebraic expressions is a fundamental skill in algebra, providing a foundation for solving equations and understanding mathematical relationships.