Problem 10 - College Algebra with Corequisite Support

Question

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies inversely as \(x\) and whe \(x=4, \quad y=2\).

Step-by-Step Solution

Answer

The equation describing the relationship is \( y = \frac{8}{x} \).

Step 1: Understand Inverse Variation

When a variable \( y \) varies inversely as another variable \( x \), it means the product of \( y \) and \( x \) is a constant. This is represented by the equation \( y = \frac{k}{x} \), where \( k \) is the constant of proportionality.

Step 2: Use Given Values to Find Constant

Substitute the given values \( x = 4 \) and \( y = 2 \) into the inverse variation equation to find the constant \( k \). Therefore, \( 2 = \frac{k}{4} \).

Step 3: Solve for the Constant of Proportionality \( k \)

Multiply both sides of the equation \( 2 = \frac{k}{4} \) by 4 to solve for \( k \). This gives \( k = 8 \).

Step 4: Write the Equation Describing the Relationship

Substitute \( k = 8 \) back into the inverse variation formula \( y = \frac{k}{x} \). The relationship is then \( y = \frac{8}{x} \).

Key Concepts

Constant of Proportionality

The constant of proportionality is a crucial concept when dealing with inverse variation. It represents a fixed number that links two variables in a specific manner.
In our scenario, when we say that "\( y \) varies inversely as \( x \)", it means that the product \( xy \) is constant for all values of \( x \) and \( y \).
This product is known as the constant of proportionality, often denoted by the letter \( k \). Think of \( k \) as a bridge keeping the relationship between \( x \) and \( y \) consistent. No matter how \( x \) changes, \( y \) will adjust so that the product \( xy \) remains equal to \( k \). In practical terms:

• If \( x \) increases, \( y \) must decrease, and vice versa.
• The constant of proportionality \( k \) provides a way to compute one variable if the other and \( k \) are known.

Inverse Variation Formula

The inverse variation formula is expressed as \( y = \frac{k}{x} \). This formula is integral to solving problems where two variables change inversely with each other.
The way it works is straightforward:

• This formula suggests that as \( x \) becomes larger, \( y \) becomes smaller.
• Conversely, as \( x \) becomes smaller, \( y \) becomes larger.

With this formula, finding \( y \) for any given \( x \) becomes a matter of simple substitution. You will already know \( k \), the constant of proportionality. In our exercise, once we determined that \( k = 8 \) based on the given values of \( x = 4 \) and \( y = 2 \), we plugged \( k \) back into the formula, thus enabling us to find \( y \) for any other \( x \) value.Remember, the inverse variation formula holds true only when the relationship is indeed inverse and \( k \) remains constant.

Algebraic Relationships

Understanding different types of algebraic relationships is key to solving real-world problems. These relationships dictate how variables behave depending on each other.
Inverse variation is one of these relationships. In its essence, it highlights how an increase in one variable leads to a proportional decrease in the other, maintaining the product as a constant. But it's just one kind:

• Direct Variation: Here, if one variable increases, the other increases too.
• Joint Variation: A scenario with more variables, direct or combined inversely.
• Combined Variation: A mix of direct and inverse variations.

An algebraic relationship like inverse variation is not always intuitive. When you work through solving problems, it helps to remember which type of variation or relationship is at play. This determines the type of formula and approach to be employed for an effective solution.