Problem 25 - Elementary Algebra

Question

Construct a mathematical model given the following. \(y\) is jointly proportional to \(x\) and \(z\), where \(y=2\) when \(x=1\) and \(z=3\).

Step-by-Step Solution

Answer

The mathematical model is \( y = \frac{2}{3}x \cdot z \).

Step 1: Define the relationship

The problem states that \( y \) is jointly proportional to \( x \) and \( z \). This means we can express \( y \) using a constant of proportionality, \( k \). The equation for joint proportionality is given by:\[ y = kx \cdot z \]

Step 2: Substitute the given values

We are given that \( y = 2 \), \( x = 1 \), and \( z = 3 \). Substitute these values into the equation to find the constant \( k \):\[ 2 = k \cdot 1 \cdot 3 \]

Step 3: Solve for the constant of proportionality

From the equation \( 2 = 3k \), solve for \( k \):\[ k = \frac{2}{3} \]

Step 4: Write the mathematical model

Now that we have found \( k \), substitute it back into the original proportional relationship to get the mathematical model:\[ y = \frac{2}{3}x \cdot z \]

Key Concepts

Mathematical Model

A mathematical model is like a recipe in mathematics. It describes the relationship between different variables using equations. In our exercise, we're asked to construct a mathematical model showing how the variable \(y\) is connected to \(x\) and \(z\). This is done through a relationship called joint proportionality.

The equation for joint proportionality combines multiplication between the variables and a constant \(k\). It looks like this:

• \(y = kx \cdot z\)

This model helps us predict what happens to \(y\) when \(x\) and \(z\) change. It is a vital tool for seeing how elements interact with each other mathematically. Understanding the model means understanding how \(y\), \(x\), and \(z\) respond when one of the factors changes.

Constant of Proportionality

The constant of proportionality, \(k\), is the hidden magic number in the world of proportions. It acts as a bridge connecting variables in an equation. When something is proportional to several other things, \(k\) helps to scale one value in terms of others, enabling accurate predictions. In our exercise, finding \(k\) occurs by using the values given: where \(y = 2\), \(x = 1\), and \(z = 3\).

• To find \(k\), you'd substitute the figures into the equation: \(2 = k \cdot 1 \cdot 3\).
• Solve this to get \(k = \frac{2}{3}\).

With \(k\) known, it becomes a fixed point in the equation, making the relationship between \(y\), \(x\), and \(z\) precise. This constant is crucial as it tells us how much \(y\) changes for any given changes in \(x\) and \(z\).

Solving Equations

Solving equations is about finding the unknowns that make an equation true. In the context of our problem, this involves finding the constant \(k\) and formulating the entire mathematical model with all known pieces of information.

To solve for \(k\), you substitute in known values, simplify the equation, and solve it just like any other algebraic equation:

• Starting from \(2 = 3k\)
• Divide both sides by 3 to isolate \(k\).
• The solution is \(k = \frac{2}{3}\).

Once \(k\) is known, substitute it back into the joint proportionality equation to finish creating the mathematical model. The equation then becomes \(y = \frac{2}{3}x \cdot z\).

This solving process allows us to use the model to predict how changing \(x\) or \(z\) would affect \(y\). Equation solving bridges the gap between conditions given in a problem and the mathematical model that represents the solution.