Problem 6 - Intermediate Algebra

Question

The equation \(y=k x\) defines ___________ variation: As \(x\) increases, y ________.

Step-by-Step Solution

Answer

The equation defines direct variation: As \(x\) increases, \(y\) increases (if \(k > 0\)).

Step 1: Identify the Type of Variation

The equation \(y = kx\) is in the form of direct variation. Direct variation means that \(y\) varies directly as \(x\), which means that when the independent variable \(x\) changes, \(y\) changes in direct proportion to \(x\).

Step 2: Understand the Relationship

In the equation \(y = kx\), \(k\) is a constant of proportionality. This means for a constant \(k\), whenever \(x\) increases, \(y = kx\) will also increase proportionally, assuming \(k > 0\). If \(k < 0\), then \(y\) will decrease as \(x\) increases.

Step 3: Fill in the Blanks

Based on the type of variation and the relationship, the blanks can be filled. The equation \(y = kx\) defines direct variation. As \(x\) increases, \(y\) increases if \(k > 0\).

Key Concepts

Constant of Proportionality

In the context of direct variation, the constant of proportionality plays a crucial role. It is represented by the letter \( k \) in the equation \( y = kx \). This constant determines how \( y \), the dependent variable, changes in relationship to \( x \), the independent variable.

The constant of proportionality \( k \) can be thought of as a multiplier. For example:

• If \( k = 2 \), then for every unit increase in \( x \), \( y \) increases by 2 units.
• If \( k = 0.5 \), the increase in \( y \) is half of the increase in \( x \).

Furthermore, \( k \) can be positive or negative:

• If \( k > 0 \), \( y \) will increase as \( x \) increases.
• If \( k < 0 \), \( y \) will decrease as \( x \) increases.

This relationship helps students predict changes in \( y \) based on changes in \( x \) depending on the value of \( k \).

Proportional Relationships

Proportional relationships are foundational in understanding direct variation. A proportional relationship between two quantities means they increase or decrease at the same rate.

In the equation \( y = kx \), this relationship is evident: as \( x \) increases, \( y \) increases proportionally by the factor \( k \). This can be visualized as a straight line when plotted on a graph, originating from the origin (0,0). The slope of this line is \( k \), indicating the rate of change.

• The line represents how \( y \) changes with \( x \) consistently.
• Being a straight line through the origin means all points \((x, y)\) satisfy the relationship \( y = kx \).
• The simplification of dividing \( y \) by \( x \) should always equal \( k \), maintaining the constancy in the relationship.

Thus, understanding proportional relationships enhances comprehension of linear functions and their graphical representation.

Variation Equations

Variation equations are mathematical statements showing how two variables relate to each other. In the case of direct variation, the equation form \( y = kx \) is used to illustrate this specific relationship.

Direct variation is one type of variation equation. It signifies that two quantities change in unison in a specific way dictated by the constant of proportionality \( k \).

• The equation \( y = kx \) clearly defines how changes in one variable, \( x \), affect the other, \( y \).
• Knowing \( k \) helps predict and calculate the outcomes when \( x \) changes. For instance, if \( x \) doubles and \( k = 3 \), then \( y \) will also double to become \( 6 \).

Variation equations offer a structured method to understand the relationship dynamics between different mathematical quantities, making them vital for solving and predicting real-world scenarios.