Problem 8 - Algebra 2

Question

Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations. $$ \begin{array}{|c|c|c|c|c|}\hline x & {3} & {5} & {7} & {10.5} \\ \hline y & {14} & {8.4} & {6} & {4} \\ \hline\end{array} $$

Step-by-Step Solution

Answer

The relationship between the values in the table is an inverse variation. The equation that models this variation is \(y = 42/x\).

Step 1: Identify the Type of Variation

Check if the product or ratio of each \(x, y\) pair is constant. A constant product indicates inverse variation, while a constant ratio indicates direct variation. If neither product nor ratio is constant, then it's neither direct nor inverse variation. For the table: \(x\) values: 3, 5, 7, 10.5 \(y\) values: 14, 8.4, 6, 4 Check the product for each pair: \(3 * 14 = 42\) \(5 * 8.4 = 42\) \(7 * 6 = 42\) \(10.5 * 4 = 42\) The products are all 42, so the relation is an inverse variation.

Step 2: Write the Equation for Inverse Variation

The equation of inverse variation is in the form \(y = k/x\), where \(k\) is the constant (in this case, 42). So, our equation based on the given values is \(y = 42/x\).

Key Concepts

Direct Variation

Direct variation describes a specific kind of mathematical relationship where two variables are proportionately linked. This means that as one variable increases, the other also increases at a constant rate. The relationship can be expressed by the equation \( y = kx \), where \( k \) is the constant of variation. Think of \( k \) like the driving force that determines how strongly \( x \) and \( y \) are linked.

Key things to remember about direct variation include:

• A straight line through the origin when plotted on a graph.
• \( k \) remains the same regardless of the specific values of \( x \) and \( y \) (as long as the variation holds).
• This concept is fundamental in understanding proportional relationships which can be found in real-world situations like speed, where distance increases directly with time if speed is constant.

In our exercise, as the expression \( y = kx \) does not fit the data (since the ratio \( \frac{y}{x} \) is not constant), the relationship is not a direct variation.

Constant Product

A constant product is a signature trait of inverse variation. It means that as one variable increases, the other decreases in such a way that their product remains unchanged. In simpler terms, when you multiply the variables together, the result is the same, no matter what values you choose. This product is denoted as \( k \), which remains consistent across all pairs of values.

To identify an inverse variation, look for a constant product in the data:

• The equation formula of inverse variation is \( y = \frac{k}{x} \), where \( k \) is the constant product.
• If the product \( x \times y = k \) holds true for all observed pairs, inverse variation is confirmed.

In the given exercise, the products were: \( 3 \times 14 = 42 \), \( 5 \times 8.4 = 42 \), \( 7 \times 6 = 42 \), and \( 10.5 \times 4 = 42 \). Here, \( k = 42 \) remains the same, indicating an inverse variation. Hence, the relationship between \( x \) and \( y \) can be expressed as \( y = \frac{42}{x} \).

Mathematical Modeling

Mathematical modeling involves using equations and formulas to represent real-world processes and relationships. It is a vital tool in mathematics, allowing us to predict behaviors and outcomes based on given data.

When faced with a set of data, like the numbers in our exercise, mathematical modeling helps in deciding whether relationships such as direct or inverse variation exist:

• The first step is analyzing data patterns (constant ratio for direct variation or constant product for inverse variation).
• Once identified, formulating the right equation accurately reflects the relationship.
• This step-by-step process mimics how scientific predictions or economic forecasts may be derived, showcasing the real-world applicability of these methods.

In our case, using the table of values, we determined the relationship through the consistent product of 42, thus modeling it via inverse variation with the equation \( y = \frac{42}{x} \). This shows how mathematical modeling helps structure and understand complex interactions in data.

Algebraic Equations

Algebraic equations are the foundation of expressing mathematical relationships between variables. They provide a concise way to describe situations where numbers or quantities are involved using letters and symbols.

The importance of algebraic equations is vast and includes:

• Helping identify relationships such as direct and inverse variations.
• Aiding in solving real-world problems by setting them up in a form that is easier to manipulate and understand.
• Allowing prediction and inference by drawing connections between known values and desired unknowns.

In our exercise, the algebraic equation \( y = \frac{42}{x} \) succinctly captures the relationship between \( x \) and \( y \) through inverse variation. Equations like this are tools that provide clarity and precision in understanding the behavior of variables together. This practice is what gives algebra its power across countless mathematical and real-world applications.