Question

Choose the best description for the model \(y=\frac{k x}{z},\) if \(k\) is a nonzero constant. (a) \(y\) varies directly with \(x\) and \(z\). (b) \(y\) is inversely proportional to \(x\) and \(z\). (c) \(y\) varies directly with \(x\) and inversely with \(z\). (d) \(y\) is directly proportion to \(z\) and inversely proportional to \(x\).

Short answer

The best description is (c).

Step-by-step solution

Understand the Equation

The given equation is \( y = \frac{kx}{z} \). Here, \(k\) is a nonzero constant.

Identify Direct Variation

In a direct variation, a variable increases if the other variable increases, and it can be expressed as \(y = kx\). Examine if there is a direct variation between \( y \) and \( x \).

Identify Inverse Variation

In an inverse variation, a variable increases if the other variable decreases, and it can be expressed as \( y = \frac{k}{z} \). Examine if there is an inverse variation between \( y \) and \( z \).

Analyze the Variation

Since \( y \) is in the numerator with \( x \), it varies directly with \( x \). Since \( y \) is in the denominator with \( z \), it varies inversely with \( z \).

Make the Conclusion

Based on the analysis, we conclude that the correct description is \( y \) varies directly with \( x \) and inversely with \( z \).

Choose the Correct Answer

Based on the analysis, the correct option is (c).

Key concepts

Direct Variation

In algebra, direct variation means that two variables change in the same direction. That is, if one variable increases, the other also increases. The relationship can be written as: \( y = kx \) where \( k \) is a constant.
Let's understand this better:

• If \( x \) doubles, then \( y \) also doubles.
• If \( x \) is halved, then \( y \) is also halved.

This constant ratio \( k \) is the key. For example, if \( k = 2 \) and you have \( x = 5 \), then \( y = 2 \times 5 = 10 \).
Direct variation can be seen in the equation from the exercise. When looking at \( y = \frac{kx}{z} \), \( y \) is directly proportional to \( x \) because if you increase \( x \), \( y \) will also increase, assuming \( z \) remains constant.

Inverse Variation

Inverse variation describes a situation where one variable increases as the other one decreases. The relationship can be expressed as: \( y = \frac{k}{z} \) where \( k \) is a constant.
Key points to consider in inverse variation:

• If \( z \) doubles, \( y \) is halved.
• If \( z \) is halved, \( y \) is doubled.

You can think of this as a see-saw effect—one side goes up while the other side goes down.
In the provided equation \( y = \frac{kx}{z} \), you can see that \( y \) varies inversely with \( z \). This means that if \( z \) increases, \( y \) will decrease and vice versa, provided \( x \) is kept constant.
This inverse relationship is crucial in solving the exercise and choosing the correct description of the model.

Proportional Relationships

Proportional relationships define how one variable changes in relation to another. These relationships can be either direct or inverse.

1. **Direct Proportionality:** As discussed before, in direct proportionality, if one variable increases, the other increases by the same factor. The equation is: \( y = kx \). This means \( y \) is directly proportional to \( x \).

2. **Inverse Proportionality:** In these relationships, one variable increases as the other decreases. The primary equation here is: \( y = \frac{k}{z} \). This indicates that \( y \) is inversely proportional to \( z \).

In the given exercise, the equation \( y = \frac{kx}{z} \) shows both types of relationships.

• \( y \) is directly proportional to \( x \) because as \( x \) increases, \( y \) also increases, assuming \( z \) stays the same.
• \( y \) is inversely proportional to \( z \) because as \( z \) increases, \( y \) decreases (with \( x \) being constant).

Understanding these principles is essential for solving problems involving variations in algebra.