Problem 3 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Variation Type Based on Direct Proportionality Formula

The formula \(y = k x z\) describes a type of variation where \(y\) changes as both \(x\) and \(z\) change. In this scenario, \(y\) is directly proportional to both \(x\) and \(z\), with \(k\) being the constant of proportionality. Thus, this is known as **joint variation**.

Step 2: Identify the Variation Type Based on Inverse Proportionality Formula

The formula \(y = \frac{k z}{x}\) involves \(y\) being inversely proportional to \(x\) because as \(x\) increases, \(y\) decreases when \(z\) and \(k\) remain constant. \(y\) is directly proportional to \(z\) and inversely proportional to \(x\), hence this is a **combined variation**.

Key Concepts

Joint Variation

Joint variation happens when one variable depends on two or more other variables together. Imagine you have a balloon that grows both in width and height. The size of the balloon (let's call it \(y\)) increases when you add air by either inflating it (\(x\)) wider or taller (\(z\)).

When this growing happens dependently, we use the expression \(y = kxz\). In this equation:

• \(y\) is what we want to understand. How much has it changed?
• \(x\) and \(z\) are the influences. They both drive the changes in \(y\).
• \(k\) is the constant of proportionality, which will be the same no matter what values \(x\) and \(z\) take on, thus giving us a steady relationship.

Joint variation is like a team effort. More of any factor corresponds to more of the outcome. If you double both \(x\) and \(z\), \(y\) quadruples, showing exactly how joint variation works.

Inverse Proportionality

Inverse proportionality is when one value increases as another one decreases. Think of it like balancing on a seesaw: if one side goes up, the other must come down. We use the formula \(y = \frac{kz}{x}\) to describe this relationship. Here:

• \(y\) will do the opposite of what \(x\) does.
• If \(x\) increases, \(y\) decreases.
• The factor \(z\) is a direct multiplier, affecting \(y\) in a straightforward way.
• The constant \(k\) ensures the seesaw follows this rule steadily over time.

This kind of variation is crucial in many real-world situations - like understanding how speed and travel time are related. Faster travel means less time on the road, showcasing inverse proportionality perfectly.

Constant of Proportionality

The constant of proportionality \(k\) is a key player in any variation equation. It's like the rule book that defines exactly how variables relate to each other.

Whether dealing with joint or inverse variation, \(k\) lets you know how much one change affects another. For instance:

• In joint variation \(y = kxz\), \(k\) helps determine precisely how \(y\) changes when both \(x\) and \(z\) change.
• In inverse proportionality \(y = \frac{kz}{x}\), \(k\) keeps things in balance as \(x\) and \(y\) play their oppositional game.

Remember, this constant remains unaffected by the alterations in \(x\), \(y\), and \(z\). It provides a foundation that keeps variable relationships constant and predictable under proportional change scenarios. When you solve any problem involving variation, finding \(k\) can unlock the mystery of how variables interact in your scenario.