Problem 29 - College Algebra

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 Inverse Variation

When a variable "varies inversely" with another variable, it means that their product remains constant. Therefore, if \( y \) varies inversely with \( x \), then they can be expressed as \( y \times x = k \), where \( k \) is a constant.

Step 2: Determine the Constant

Use the given values \( x = 3 \) and \( y = 2 \) to find the constant \( k \). Substitute these values into the equation \( y \times x = k \). This gives us \( 2 \times 3 = k \), so \( k = 6 \).

Step 3: Solve for the Unknown Value

Now that we know \( k = 6 \), use it to find \( y \) when \( x = 1 \). Using the equation \( y \times x = k \), substitute \( x = 1 \) and \( k = 6 \). This gives us \( y \times 1 = 6 \), therefore \( y = 6 \).

Key Concepts

Constant of Variation

In inverse variation, the constant of variation plays a crucial role. This constant is commonly denoted as \( k \). When a problem states that one variable varies inversely with another, it means the product of those variables equals a constant.
In mathematical terms, if \( y \) varies inversely with \( x \), the equation \( y \times x = k \) comes into play. Here, \( k \) is a constant number that doesn't change even if \( x \) and \( y \) do.

• Think of \( k \) as the glue holding \( x \) and \( y \) together in a fixed relationship.
• The constant of variation helps us adjust one variable when the other changes.

Understanding this constant allows us to predict how one variable behaves when the other varies. Once we determine \( k \) using initial values, we can solve for unknowns in a consistent way.

Solving Equations

Solving equations, especially in inverse variation, involves understanding the relationship between the variables. Once we know our equation (\( y \times x = k \)), tracing the unknown value becomes straightforward.
When given initial values of \( x \) and \( y \), you can find \( k \) by substituting these values into the equation. This makes finding \( k \) an essential step:

• Substitute the known values into the equation to calculate \( k \).
• Reverse engineer the problem: now that \( k \) is known, you can solve for the unknown variable using the same base equation.
• This process is consistent regardless of which variable is unknown.

Every step in solving inverse variation equations serves to maintain the integrity of the original inverse relationship as dictated by the constant \( k \).

Direct and Inverse Relationships

Understanding the difference between direct and inverse relationships is key in many mathematical problems. In a direct variation, two variables change in the same direction. If one increases, so does the other. Alternatively, inverse variation implies an opposite change:

• In direct variation: \( y = kx \), the variables increase or decrease together.
• In inverse variation: \( y \times x = k \), as one variable increases, the other decreases.

Understanding these relationships is like understanding the push and pull dynamic in different scenarios. In inverse relationships, the notion that \( y \) gets larger as \( x \) gets smaller (and vice versa) explains many natural occurrences. This concept is at the heart of exercises involving inverse variations, ensuring you grasp how variables relate to each other in diverse situations.