Problem 4 - Elementary and Intermediate Algebra

Question

The equation \(y=\frac{k x}{z}\) means that \(y\) varies _____ with \(x\) and _____ with \(z\).

Step-by-Step Solution

Answer

y varies directly with x and inversely with z.

Step 1: Analyzing the Relationship of y and x

The equation given is \( y = \frac{kx}{z} \). Notice that \( y \) is on the left side and \( x \) is in the numerator on the right-hand side. If \( k \) and \( z \) are constants, then \( y \) changes proportionally with \( x \). This relationship where \( y \) increases as \( x \) increases (and vice versa) is described as "direct variation." Thus, \( y \) varies directly with \( x \).

Step 2: Analyzing the Relationship of y and z

Continuing with the equation \( y = \frac{kx}{z} \), observe that \( z \) is in the denominator. When \( z \) increases, \( y \) decreases, assuming \( k \) and \( x \) remain constant. Conversely, when \( z \) decreases, \( y \) increases. This relationship where \( y \) changes inversely with changes in \( z \) is called "inverse variation." Therefore, \( y \) varies inversely with \( z \).

Key Concepts

Direct Variation

In mathematics, direct variation describes a relationship between two variables where one is a constant multiple of the other. Simplified, this means as one variable increases, the other increases, too, and as one decreases, the other decreases. This relationship is easily captured using the formula: \[ y = kx \] Where:

• \( y \) is the variable directly varying with \( x \)
• \( x \) is the variable we change
• \( k \) is the constant of variation or proportionality

This equation shows that \( y \) changes at a rate determined by the constant \( k \). If \( k \) is positive, both variables increase and decrease together. If \( k \) is negative, when one variable increases, the other decreases, though technically this isn't considered a direct variation by most definitions. The critical component is the direct multiplication that defines how the variables change together. Direct variation is often useful in physics and economics, such as calculating the distance traveled by an object over time with constant speed.

Inverse Variation

Inverse variation describes a situation where one variable increases while the other decreases proportionally. The mathematical expression for such a relationship is: \[ y = \frac{k}{z} \] Here:

• \( y \) is inversely varying with \( z \)
• \( z \) is the independent variable
• \( k \) is a non-zero constant, indicating the strength of the relationship

In our equation, as \( z \) becomes larger, \( y \) becomes smaller if \( k \) remains the same, and vice versa. This is why \( y \) and \( z \) show inverse behavior. This concept is crucial when dealing with situations where quantities vary inversely, such as speed and travel time for a fixed distance. As speed increases, travel time decreases proportionally, indicating an inverse relationship.

Proportional Relationships

Having a firm grasp of proportional relationships is invaluable in understanding both direct and inverse variations. A proportional relationship refers to a situation where two quantities maintain a constant ratio. This can be expressed with either direct or inverse variation equations.

• In a direct variation, the ratio between the two variables is constant as described by \( \frac{y}{x} = k \).
• In inverse variation, the product of the two variables remains constant: \( y \cdot z = k \).

Understanding proportional relationships allows you to predict and calculate how one variable responds to changes in another. These relationships are foundational in geometry and physics where they help describe how variables are interconnected. They simplify complex systems by reducing variables to constants, making problem-solving more straightforward.