Problem 19 - Elementary Algebra

Question

Construct a mathematical model given the following. \(y\) is inversely proportional to \(x,\) and \(y=3\) when \(x=9\).

Step-by-Step Solution

Answer

The model is \( y = \frac{27}{x} \).

Step 1: Understanding Inverse Proportion

When a variable is inversely proportional to another, it means their product is constant. Mathematically, if \( y \) is inversely proportional to \( x \), we can express this relationship as \( y \propto \frac{1}{x} \) or equivalently, \( y = \frac{k}{x} \), where \( k \) is a constant.

Step 2: Substitute Given Values to Find Constant

We are given \( y = 3 \) when \( x = 9 \). To find \( k \), substitute these values into the equation \( y = \frac{k}{x} \). This gives us \( 3 = \frac{k}{9} \).

Step 3: Solve for Constant \( k \)

To solve for \( k \), multiply both sides of the equation \( 3 = \frac{k}{9} \) by 9: \[ k = 3 \times 9 = 27 \]

Step 4: Write the Mathematical Model

With \( k \) found to be 27, substitute it back into the expression for the inverse proportion: \[ y = \frac{27}{x} \] This is the mathematical model that describes the relationship between \( y \) and \( x \).

Key Concepts

Mathematical Model

In mathematics, a model is essentially an equation or a set of equations that represent a real-world situation. It helps us to understand and analyze relationships between different variables. In the context of inverse proportion, a mathematical model illustrates how one variable changes when another variable changes.
When you read a problem stating that \( y \) is inversely proportional to \( x \), it implies that as one variable increases, the other decreases. They move in opposite directions.
The specific form of the mathematical model for inverse proportion is \( y = \frac{k}{x} \). Here, \( k \) is a constant that remains the same no matter what values \( x \) and \( y \) take, as long as they maintain the inverse relationship. This model helps one predict how changes in \( x \) will affect \( y \), by substituting any value of \( x \) into the equation.

• The model represents the core idea of inverse relationships.
• It translates the real-world scenario of inverse proportionality into mathematical terms.
• The equation \( y = \frac{k}{x} \) allows us to calculate the value of one variable if we know the other and the constant \( k \).

Constant of Proportionality

The constant of proportionality, denoted by \( k \) in the equation \( y = \frac{k}{x} \), is an essential part of describing inverse relationships. It defines the strength or magnitude of the relationship between two inversely proportional variables.
In the given example, \( y = 3 \) when \( x = 9 \), we use this information to find \( k \). By substituting into the equation, \( 3 = \frac{k}{9} \), we solve for \( k \). Multiplying both sides by 9, we find \( k = 27 \).
This value of \( k \) tells us that for each unit change in \( x \), there is a fixed impact on \( y \), dictated by this constant. It's a fixed anchor point in an otherwise dynamic relationship.

• \( k \) remains consistent regardless of changes in \( x \) or \( y \).
• It determines the steepness or rate of change in the inverse relationship.
• Once \( k \) is known, any value of \( x \) can predict \( y \) using the inverse proportion model.

Algebraic Equation

An algebraic equation is a statement of equality involving variables and constants. It is used to express relationships and solve for unknown variables. In our inverse proportion scenario, the algebraic equation is \( y = \frac{k}{x} \). This equation tells us how \( y \) varies with \( x \).
Algebraic equations are foundational in math as they provide clarity and structure to problem-solving. They distill complex relationships into simple, manipulable forms.
In the given exercise, after determining \( k \), the full equation \( y = \frac{27}{x} \) becomes the algebraic tool we use to analyze and infer outcomes.

• This equation is solved typically through substitution and solving steps.
• The algebraic method requires determining unknown constants like \( k \).
• Once formed, it aids in drawing further conclusions about the variable behavior.