Problem 8 - Intermediate Algebra

Question

The equation \(y=k x z\) defines ___________ variation, and \(y=\frac{k z}{x}\) defines __________ variation.

Step-by-Step Solution

Answer

Direct variation; Inverse variation.

Step 1: Identify the Type of Variation for the First Equation

The equation given is \(y = kxz\). In this scenario, \(y\) varies directly with \(x\) and \(z\). Direct variation means as \(x\) or \(z\) increase, \(y\) increases as well and vice versa. Therefore, this equation defines a direct variation between \(y\), \(x\), and \(z\).

Step 2: Identify the Type of Variation for the Second Equation

The second equation is \(y = \frac{kz}{x}\). Here, \(y\) varies directly with \(z\) but inversely with \(x\). Inverse variation means that as \(x\) increases, \(y\) decreases and vice versa, assuming \(z\) and \(k\) stay constant. Thus, this equation defines an inverse variation between \(y\) and \(x\).

Step 3: Conclusion of Both Equations

Based on the analysis above: the first equation \(y=kxz\) describes a direct variation, and the second equation \(y=\frac{kz}{x}\) describes an inverse variation.

Key Concepts

Direct Variation

Direct variation occurs when one quantity increases or decreases in direct proportion to another quantity. It's a straightforward concept that can be expressed with a simple algebraic equation:

• The formula used for direct variation is usually represented as \( y = kx \), where \( y \) and \( x \) are the variables, and \( k \) is the constant of variation.
• This means that \( y \) changes at a constant rate of \( k \) for every unit change in \( x \).
• If you double \( x \), then \( y \) also doubles, assuming \( k \) is constant.

Let's relate this back to the given equation \( y = kxz \). In this equation, \( y \) varies directly with both \( x \) and \( z \). This means if either \( x \) or \( z \) increases, \( y \) will increase proportionally, given the other variable remains constant.

Inverse Variation

Inverse variation describes a situation where one quantity increases as another quantity decreases. This type of variation is common in scenarios where a product is constant. It is represented by the formula:

• \( y = \frac{k}{x} \), where \( y \) decreases as \( x \) increases if \( k \) is positive.
• The constant \( k \) represents the product \( y \times x \), which must remain constant.
• As \( x \) gets larger, \( y \) must get smaller to maintain the value of \( k \).

In the equation \( y = \frac{kz}{x} \), \( y \) varies inversely with \( x \). This means that if \( x \) increases, \( y \) decreases, as long as \( z \) and \( k \) stay constant. This inverse relationship is crucial for understanding many real-world problems where balance or equilibrium is necessary.

Algebraic Equations

Algebraic equations are fundamental tools in mathematics used to describe relationships between different variables. Understanding them involves knowing the parts of an equation:

• Variables represent unknown values and are often noted as letters such as \( x \), \( y \), or \( z \).
• Coefficients like \( k \) define the rate or degree of variation between variables.
• Operations such as multiplication and division show how variables interact with each other.

In the example given, the equations \( y = kxz \) and \( y = \frac{kz}{x} \) are both algebraic, each expressing a unique type of relationship:- **Direct variation** in the first equation shows a multiplicative relationship among the variables.- **Inverse variation** in the second reflects a division operation, highlighting a dependency where increasing one variable leads to a decrease in another.Algebraic equations like these are powerful for modeling real-world situations, helping solve practical problems effectively.