Problem 20 - Algebra 2

Question

State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation. \(p=\frac{12}{q}\)

Step-by-Step Solution

Answer

It represents an inverse variation; the constant of variation is 12.

Step 1: Identify the Form of the Equation

An inverse variation equation is usually represented in the form \(y = \frac{k}{x}\), where \(k\) is the constant of variation. Compare \(p = \frac{12}{q}\) to this standard form.

Step 2: Determine the Type of Variation

In the equation \(p = \frac{12}{q}\), observe that as \(q\) increases, \(p\) decreases and vice versa. This fits the description of an inverse variation equation because \(p\) is inversely proportional to \(q\).

Step 3: Identify the Constant of Variation

In the inverse variation \(p = \frac{k}{q}\), the constant of variation \(k\) can be directly identified from the equation. Here, \(k = 12\) because \(p\) is expressed as \(\frac{12}{q}\).

Key Concepts

Constant of Variation

In algebra, the **constant of variation** is essential because it plays a crucial role in equations representing proportional relationships. It is the fixed value, noted as \( k \) in an equation, that defines how two variables correlate with each other. In our example, we have the inverse variation equation \( p = \frac{12}{q} \). Here, the constant of variation is palpable in the numerator. You can see that \( k = 12 \).

• The constant of variation represents a fixed number in these mathematical relationships.
• It signifies the rate at which one variable changes with another.
• In inverse variation, it shows the product of the two variables.

Recognizing the constant helps us understand the inherent relationship between the variables. It lets us compute values one variable can take given another.

Proportional Relationships

Proportional relationships describe how two quantities change together. There are typically three forms:

• Direct Variation: \( y = kx \)
• Inverse Variation: \( y = \frac{k}{x} \)
• Joint Variation: Sometimes involves more than two variables.

In inverse variation, when one variable increases, the other decreases, maintaining the product constant (\( k \)). Using our example \( p = \frac{12}{q} \), as \( q \) goes up, \( p \) must go down so that their product remains equal to 12. This reciprocal relationship is inverse proportionality.
Understanding these types of relationships helps to solve and predict scenarios that involve changing variables. Knowing whether variables are directly, inversely, or jointly proportional guides our problem-solving approach.

Mathematical Equations

Equations are the bedrock of expressing mathematical ideas and relationships. Each element in an equation, including functions and constants, tells a story. For instance, the equation \( p = \frac{12}{q} \) tells us how \( p \) depends on \( q \) through an inverse variation. Here:

• \( p \) is dependent upon \( q \).
• 12 is the constant of variation.
• \( q \)'s behavior inversely affects \( p \).

Equations enable us to model real-world phenomena and allow us to make predictions. Whether solving for unknown variables or graphing relationships, equations utilize constants of variation and proportionality as foundational concepts.
These mathematical tools make the abstract concrete, turning dynamic and complex interactions into manageable calculations. Properly understanding them opens the door to more advanced mathematical or scientific concepts.