Problem 28 - Intermediate Algebra

Question

If \(y\) is inversely proportional to \(x\), and \(y=\frac{1}{35}\) when \(x=14\), find the value of \(y\) when \(x=16\).

Step-by-Step Solution

Answer

When \( x = 16 \), the value of \( y \) is \( \frac{1}{40} \).

Step 1: Understand the Relationship

If \( y \) is inversely proportional to \( x \), then we can express the relationship as \( y = \frac{k}{x} \) where \( k \) is a constant. Our goal is to find the constant \( k \) using the given values.

Step 2: Find the Constant k

Use the given values \( y = \frac{1}{35} \) and \( x = 14 \) to find \( k \). Substitute into the equation: \( \frac{1}{35} = \frac{k}{14} \). Solve for \( k \) by multiplying both sides by 14: \( k = \frac{1}{35} \times 14 = \frac{14}{35} = \frac{2}{5} \).

Step 3: Calculate y for New x Value

Now that we have \( k = \frac{2}{5} \), find \( y \) when \( x = 16 \). Plug in the values into the equation \( y = \frac{k}{x} \): \( y = \frac{\frac{2}{5}}{16} = \frac{2}{5} \times \frac{1}{16} = \frac{2}{5 \times 16} = \frac{2}{80} = \frac{1}{40} \).

Key Concepts

Algebraic Relationship

When two quantities have a special link represented through an algebraic expression, we refer to this as an algebraic relationship. In our case, the quantities are inversely related. This means as one quantity increases, the other decreases. The mathematical expression that captures this relationship is given by \( y = \frac{k}{x} \), where \( k \) is a constant and \( y \) is inversely proportional to \( x \).
Understanding this equation is crucial because it shows how changes in \( x \) lead to changes in \( y \).
Inverse proportionality is different from direct proportionality, which is when two quantities increase or decrease together. In inverse relationships, it's exactly the opposite. So for any increase in \( x \), \( y \) will decrease by a factor proportional to the change in \( x \), and vice versa. This defining characteristic is what makes the algebraic relationship useful for solving problems involving inverse proportions.

Proportionality Constant

The constant of proportionality, \( k \), is a crucial element in understanding inverse variation. This constant makes it possible to quantify how much one variable affects another. In the relationship \( y = \frac{k}{x} \), \( k \) keeps the equation balanced no matter what values \( x \) takes on.
The constant can be determined using a set of known values for \( x \) and \( y \). By substituting these known values into the equation, \( k \) can be isolated and calculated. For example, if we know \( y = \frac{1}{35} \) when \( x = 14 \), this information can be used to find \( k \).
Finding \( k \) involves rearranging the equation so that \( k \) is the subject, allowing you to solve for it easily. Knowing this constant lets us make accurate predictions for other values of \( x \) and \( y \) within this relationship.

Solving Equations

To solve an equation involving inverse proportionality, you need to follow a clear step-by-step approach. First, identify the known values of \( x \) and \( y \), and substitute them into \( y = \frac{k}{x} \). This helps derive the proportionality constant \( k \).
Once you have \( k \), you can then substitute this back into the equation to solve for new values of \( y \) or \( x \). For instance, after determining \( k = \frac{2}{5} \), we calculated \( y \) for a new \( x = 16 \) using the same formula:

• Substitute: \( y = \frac{\frac{2}{5}}{16} \)
• Simplify the fraction: \( y = \frac{2}{80} \)
• Result: \( y = \frac{1}{40} \)

Each step involves logical manipulations based on algebraic principles to arrive at the answer. Understanding how to efficiently solve these equations by substituting and simplifying can help determine unknown variables in various inverse relationships.