Problem 91 - College Algebra with Modeling and Visualization

Question

Find the constant of proportionality \(k\) $$ y=\frac{k}{x}, \text { and } y=2 \text { when } x=3 $$

Step-by-Step Solution

Answer

The constant of proportionality \(k\) is 6.

Step 1: Understand the Relationship

We are given that the relationship between two variables, \(y\) and \(x\), is defined by the equation \(y = \frac{k}{x}\). This is an inverse variation, where \(k\) is the constant of proportionality.

Step 2: Substitute Given Values

We know from the problem that \(y = 2\) when \(x = 3\). Substitute these values into the equation: \(2 = \frac{k}{3}\).

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

To find \(k\), we need to isolate it on one side of the equation. Multiply both sides of the equation \(2 = \frac{k}{3}\) by 3 to get: \[ k = 2 imes 3 \]

Step 4: Simplify the Expression

Calculate \(k\): \[ k = 6 \] The constant of proportionality is 6.

Key Concepts

Understanding the Constant of Proportionality

The constant of proportionality, often represented by the letter "k," is a crucial component in relationships where two variables are involved. In inverse variation equations like \( y = \frac{k}{x} \), the constant of proportionality represents how one variable changes in response to another. Here, the value of \( k \) indicates how strongly \( y \) is affected by changes in \( x \).
Inverse variation differs from direct variation. In direct variation, both variables change in the same direction. However, in inverse variation, while one variable increases, the other decreases, ensuring their product remains constant.
A real-world analogy could be filling a tank of water. The speed of filling can depend inversely on the pipe’s width. The constant represents the total water needed, so with a wider pipe, less time is needed and vice-versa. The constant doesn’t change as it depends on the setup itself, not the variables.

• Remembering it in terms of a real setting can help understand its unchanged nature even as variables shift.

How to Substitute Values Correctly

Substituting values in math equations is a straightforward process, essential for solving equations accurately. It involves replacing variables with the given numerical values and simplifying the expression. Let’s consider our equation: \( y = \frac{k}{x} \).
We know from the problem statement, \( y = 2 \) and \( x = 3 \). By substituting \( x \) and \( y \) with these values, the equation becomes \( 2 = \frac{k}{3} \).
This substitution helps transform an equation with variables into one that we can solve directly since it becomes purely numerical.
Double-check substitutions to ensure accuracy. Mistaking one value for another can lead to incorrect results.

• Make a habit of laying out known values before substituting. It keeps the process orderly and helps avoid missteps.
• For clarity, write the original equation, then rewrite it after substitution.

Steps for Solving Equations

Solving equations by isolating variables is a fundamental skill in algebra. After substituting known values into our equation, we proceed by isolating the constant or desired variable. In the equation \( 2 = \frac{k}{3} \), we aim to solve for \( k \).
The key is to perform the same operation on both sides of the equation—this maintains equality. Here, multiplying both sides by 3 eliminates the fraction, leaving \( k = 2 \times 3 \).
Multiplication here helps "move" the denominator (3) from one side, clarifying the value on the other side. The operation must "balance" across the equation.
After operations, the final step is to calculate the remaining expression, providing \( k = 6 \) in this example.

• It's often helpful to write down each step to track the equation’s changes.
• Perform a quick mental check with the original equation to ensure results make sense.
• If an operation doesn’t "work" or alters the equality, consider rethinking the problem or checking earlier steps for mistakes.