Problem 15 - Elementary and Intermediate Algebra

Question

Complete each solution. Determine whether the equation defines direct variation. a. \(y=k x\) b. \(y=k+x\) c. \(y=\frac{k}{x}\) d. \(m=k c\)

Step-by-Step Solution

Answer

Equations (a) and (d) define direct variation; (b) and (c) do not.

Step 1: Definition of Direct Variation

An equation represents a direct variation if it can be written in the form of \(y = kx\), where \(k\) is a constant and \(x\) is the variable. This means that \(y\) changes directly with \(x\) such that if \(x\) increases, \(y\) increases and vice versa.

Step 2: Analyze Equation (a)

Equation (a) is \(y = kx\). This matches the form of the direct variation equation \(y = kx\). Thus, the equation defines a direct variation.

Step 3: Analyze Equation (b)

Equation (b) is \(y = k + x\). This does not match the form \(y = kx\) because of the addition of \(k\) instead of a multiplication. Thus, the equation does not define a direct variation.

Step 4: Analyze Equation (c)

Equation (c) is \(y = \frac{k}{x}\). This does not match the form \(y = kx\), because it involves division by \(x\) instead of multiplication with \(x\). Hence, the equation does not define a direct variation.

Step 5: Analyze Equation (d)

Equation (d) is \(m = kc\). This matches the form of direct variation \(y = kx\), with \(m\) as the dependent variable, \(c\) as the independent variable, and \(k\) as the constant. Therefore, the equation defines a direct variation.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements composed of variables and constants. These equations often illustrate various relationships and patterns observed in mathematics. Direct variation, a specific type of algebraic equation, involves a relationship where one variable changes directly as another variable. In a direct variation, the equation is typically expressed in the form \( y = kx \), where \( y \) and \( x \) are variables and \( k \) is a constant called the constant of variation.

When studying algebraic equations, it's essential to understand the operations involved:

• Addition and subtraction change values linearly.
• Multiplication and division influence how variables and constants scale with each other.
• Equations like \( y = kx \) express these multiplicative relationships.

This knowledge helps in analyzing and solving different types of equations, including those involving direct variation, making algebra a powerful tool in mathematical problem-solving.

Constant and Variable Relationships

In mathematics, constants and variables play critical roles in defining relationships within equations. Constants are fixed values, while variables can change or vary, accommodating various inputs and outputs. In a direct variation, the relationship is between a constant \( k \) and two variables, often written as \( y = kx \).

This simple formula helps us understand:

• How changes in one variable affect another variable. For example, if \( k = 3 \) and \( x = 4 \), then \( y = 3\times4 = 12 \). This shows that when \( x \) increases, \( y \) increases proportionally.
• The proportional relationship: No matter the values of \( x \) and \( y \), the ratio \( \frac{y}{x} \) remains constant, equal to \( k \).

Such relationships are fundamental for understanding other mathematical concepts, demonstrating how structured and predictable mathematics can be.

Mathematical Analysis

Mathematical analysis involves examining equations and their components to identify underlying relationships. In the case of direct variation, analysis focuses on determining if changes in one variable consistently lead to proportional changes in another variable. This is verified by ensuring equations conform to the standard form \( y = kx \).

Consider the equations provided in the exercise:

• Equation (a), \( y = kx \), directly fits this form, confirming it demonstrates direct variation.
• Equations (b) and (c), \( y = k + x \) and \( y = \frac{k}{x} \), don’t conform to this direct form. Thus, they do not exhibit direct variation.
• Equation (d), \( m = kc \), is similar to \( y = kx \) but with different variable names, confirming a direct variation relationship.

Through analysis, students can identify and categorize mathematical relationships, enhancing their comprehension of algebraic concepts.