Problem 75 - Elementary Algebra

Question

Construct a mathematical model given the following. $$ y \text { varies directly with } x \text { , and } y=12 \text { when } x=4 $$

Step-by-Step Solution

Answer

The mathematical model is \( y = 3x \).

Step 1: Understanding Direct Variation

Direct variation is a relationship between two variables in which one is a constant multiple of the other. When we say that \(y\) varies directly with \(x\), it means that \[ y = k imes x \] where \(k\) is a constant called the constant of variation.

Step 2: Setting Up the Relationship

From the variation description, we write the equation \[ y = k imes x \] where \(k\) is the constant we need to find. We are given that when \(x = 4\), \(y = 12\).

Step 3: Solving for the Constant of Variation

Substitute the given values of \(x\) and \(y\) into the equation: \[ 12 = k imes 4 \] Then divide both sides of the equation by 4 to solve for \(k\): \[ k = \frac{12}{4} = 3 \]

Step 4: Constructing the Mathematical Model

Now that we have found \(k = 3\), substitute back into the direct variation formula to construct the mathematical model: \[ y = 3x \]

Key Concepts

Constant of Variation

The concept of the constant of variation is fundamental when discussing direct variation between two variables. When we say that a variable \( y \) varies directly with another variable \( x \), it implies a consistent proportional relationship between the two. This is expressed mathematically as \( y = k \times x \), where \( k \) is the constant of variation.

• The constant of variation \( k \) represents the rate at which \( y \) changes relative to changes in \( x \).
• In our exercise, when \( x = 4 \), \( y = 12 \), allowing us to calculate \( k \) as \( 3 \).
• Thus, \( k = \frac{12}{4} = 3 \).

This constant remains the same for any pair of values \( x \) and \( y \) that satisfy the relationship; meaning \( y \) will always be three times the value of \( x \) in this case. This consistent ratio is what defines the direct variation.

Mathematical Model

Creating a mathematical model involves translating real-world relationships into mathematical language. Here, the problem describes a direct variation between \( y \) and \( x \), which lends itself to a straightforward mathematical model: \( y = k \times x \).

• From the given data (\( y = 12 \) when \( x = 4 \)), find \( k \) by solving the equation \( 12 = k \times 4 \), resulting in \( k = 3 \).
• Substitute \( k = 3 \) back into the equation, giving us the model \( y = 3x \).

This equation now serves as a predictive tool or a model to describe how \( y \) behaves related to any value of \( x \). Mathematical models like this one simplify complex relationships, allowing for predictions and understanding of how changing one variable impacts the other in direct variation scenarios.

Linear Equations

Linear equations are critical in understanding direct variation relationships. A linear equation in its basic form is \( y = mx + b \). However, in direct variation, it simplifies to \( y = kx \) because the y-intercept \( b \) is zero. This is why direct variation is a specific type of linear equation that passes through the origin.

• The equation \( y = 3x \) is linear because it graphs a straight line through the origin with a slope of \( 3 \).
• The slope, \( 3 \), indicates the rate at which \( y \) increases as \( x \) increases.

Understanding linear equations with direct variation helps visualize the constant proportional change between variables. The graph of such an equation will always be a straight line, reflecting the direct and constant ratio described by \( k \). This fundamental understanding enables one to interpret how direct variation impacts real-world situations.