Problem 29 - College Algebra with Corequisite Support

Question

For the following exercises, use the given information to find the unknown value. \(y\) varies inversely with \(x\). When \(x=3\), then \(y=2\). Find \(y\) when \(x=1\).

Step-by-Step Solution

Answer

When \( x = 1 \), \( y = 6 \).

Step 1: Understand the Concept of Inverse Variation

Inverse variation implies that when one value increases, the other value decreases proportionally. Mathematically, this relationship is represented as \( y = \frac{k}{x} \), where \( k \) is the constant of variation.

Step 2: Find the Constant of Variation

We are given that when \( x = 3 \), \( y = 2 \). Substitute these values into the inverse variation formula to find \( k \): \( 2 = \frac{k}{3} \). Solving for \( k \), we multiply both sides by 3 to get \( k = 6 \).

Step 3: Set Up the Equation with the Constant

Now that we know \( k = 6 \), we can express the relationship between \( x \) and \( y \) as \( y = \frac{6}{x} \).

Step 4: Find the Unknown Value of \( y \)

To find \( y \) when \( x = 1 \), substitute \( x = 1 \) into the equation: \( y = \frac{6}{1} \). This simplifies to \( y = 6 \).

Key Concepts

Constant of Variation

In inverse variation, the **constant of variation** is a key concept that helps us understand the relationship between two variables – here, between \( x \) and \( y \). In the exercise provided, \( y \) varies inversely with \( x \), meaning that as one increases, the other decreases, but always by the same proportional amount, which is determined by this constant.

• Mathematically, the constant of variation \( k \) is found using the formula \( y = \frac{k}{x} \).
• In our example, when \( x = 3 \) and \( y = 2 \), substituting these into the equation gives \( 2 = \frac{k}{3} \).
• Solving for \( k \), we assume multiplication of both sides by 3, resulting in \( k = 6 \).

This constant \( k \) tells us how tightly \( y \) clings to changes in \( x \). Once calculated, it provides a solid basis to find unknown values of \( y \) if \( x \) changes.

Proportional Relationship

In a **proportional relationship**, the concept transforms slightly when discussing inverse variation. While direct variation involves a straightforward increase or decrease between variables, inverse variation operates with a complementary dance.
In the exercise, when \( y \) varies inversely with \( x \), it tells us:

• As \( x \) increases, \( y \) decreases, and vice-versa.
• The product \( xy \) remains constant because of the relationship \( xy = k \).

Let's revisit the example to see the correlation. Given \( y = \frac{k}{x} \) and knowing \( k = 6 \), the product \( xy = 6 \) remains constant. This consistency is what characterizes the proportional relationship in inverse variation; despite the inverse nature, there's a steadiness governed by the unwavering constant \( k \).

Algebraic Equations

Algebraic equations are the backbone of solving problems involving inverse variations, allowing us to derive and compute unknown values effortlessly. The process engages our problem-solving skills and arithmetic manipulations.
Here's how we can think about it:

• Start with the basic inverse variation formula: \( y = \frac{k}{x} \).
• Substitute the known values to solve for \( k \), i.e., \( 2 = \frac{k}{3} \).
• Rearrange (if needed) and solve for the constant, \( k = 6 \), which models the relationship.

Once we have our equation with \( k \) known, it's a simple substitution task to find \( y \) for any given value of \( x \). For instance, when \( x = 1 \), put into the equation \( y = \frac{6}{1} \), giving us \( y = 6 \).
These algebraic steps streamline finding relationships in varying contexts, building a sturdy bridge from known to unknown values.