Problem 101 - College Algebra with Modeling and Visualization

Question

Let \(y\) be directly proportional to \(x\) Complete the following. Find \(y\) when \(x=5,\) if \(y=7\) when \(x=14\)

Step-by-Step Solution

Answer

When \(x = 5\), \(y = 2.5\).

Step 1: Understanding Direct Proportionality

Since the word 'directly proportional' is used, it means that as one variable increases, the other increases at a constant rate. Mathematically, this is represented as \( y = kx \), where \(k\) is the constant of proportionality.

Step 2: Finding the Constant of Proportionality

We use the information given: \(y = 7\) when \(x = 14\). Substitute these values into the equation \(y = kx\) to find \(k\). \[ 7 = k \times 14 \]Solve for \(k\): \[ k = \frac{7}{14} = \frac{1}{2} \]

Step 3: Setting up the Proportionality Equation

Now that we have \(k\), we can express the relationship as \( y = \frac{1}{2}x \). This equation shows that \(y\) increases by \(\frac{1}{2}\) for every increase of 1 in \(x\).

Step 4: Calculating \(y\) for a Given \(x\)

We need to find \(y\) when \(x=5\). Substitute \(x=5\) into our equation \( y = \frac{1}{2}x \): \[ y = \frac{1}{2} \times 5 \]This simplifies to: \[ y = \frac{5}{2} = 2.5 \]

Step 5: Final Answer Verification

Ensure the solution matches the directly proportional relationship. We expect \(y\) to be \(2.5\) when \(x\) is 5 given our derived equation. Double-check calculations in previous steps to confirm correctness.

Key Concepts

Constant of Proportionality

The constant of proportionality, often denoted as "\( k \)," is a crucial component in understanding direct proportionality. It represents the consistent ratio between two directly proportional variables. In our exercise, we discover this constant by using the known values of \( x \) and \( y \). Here, we were given that \( y = 7 \) when \( x = 14 \). We can substitute these values into the equation of direct proportionality, \( y = kx \), which gets us \( 7 = k \times 14 \). By solving for \( k \), we find \( k = \frac{1}{2} \).

This constant \( k = \frac{1}{2} \) tells us that for every unit increase in \( x \), \( y \) increases by half. This constant does not change, reflecting the consistent rate at which \( y \) changes relative to \( x \). This concept is vital for grasping how relationships in mathematics hold steady under proportional conditions.

Mathematical Modeling

Mathematical modeling is a method of creating equations to represent real-world situations. In the given problem context, we use mathematical modeling to describe the relationship between \( x \) and \( y \) where they are directly proportional.

• Identify Variables: Recognize which quantities vary and need to be represented in a mathematical manner. Here, \( x \) and \( y \) were identified.
• Create Relationships: Use known information to write equations. By establishing that \( y \) directly depends on \( x \) with a constant rate of change, we form the equation \( y = kx \).
• Apply the Model: Insert specific values into the equation to predict outcomes or understand the relationship better.

In our case, we converted the description of direct proportionality into a mathematical statement using \( y = \frac{1}{2}x \). This model now allows us to predict \( y \) for any given \( x \), which is a powerful tool for both simple and complex problem-solving scenarios.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality of two expressions. In this exercise, the essential equation that describes the relationship between \( y \) and \( x \) is an algebraic equation derived from the principle of direct proportionality: \( y = kx \).

To solve for \( y \), we substituted the constant of proportionality \( k = \frac{1}{2} \), forming the equation \( y = \frac{1}{2}x \). When using algebraic equations:

• Balance both sides: Ensure both sides represent equal values when replaced with the actual numbers.
• Solve for unknowns: Isolate the variable you're solving for by using operations like addition, subtraction, multiplication, and division.
• Verify Solutions: Double-check the values satisfy the original equation principles.

In this task, when \( x = 5 \), we simply substituted into the equation \( y = \frac{1}{2} \times 5 \) which resulted in \( y = 2.5 \). Confirming this answer aligns with our equation verifies our algebraic and mathematical processes are correct.