To solve inverse proportionality problems using the formula $y = \frac{k}{x}$, you can follow these steps:

Process to Solve:

1. Understand the Relationship:

   • Inverse proportionality means that as $x$ increases, $y$ decreases, and vice versa. The constant $k$ is known as the constant of proportionality.

2. Identify Known Values:

   • You need at least one set of $(x, y)$ values and the constant $k$ to solve for the unknown.

3. Write the Formula:

   • The basic formula for inverse proportionality is $y = \frac{k}{x}$, where $k$ is the constant of proportionality.

4. Find the Constant $k$:

   • If you are given a pair of $(x, y)$, substitute these values into the formula to solve for $k$: $k = x \cdot y$

5. Use $k$ to Solve for Other Values:

   • After finding $k$, you can use it to calculate $y$ for different values of $x$, or solve for $x$ if $y$ is known: $y = \frac{k}{x} \quad \text{or} \quad x = \frac{k}{y}$

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Flashcard Example:

Q: How do you solve inverse proportionality using the formula $y = \frac{k}{x}$?

A:

1. Recognize the relationship $y = \frac{k}{x}$.
2. Identify given values for $x$ and $y$.
3. Find the constant $k$ by multiplying $x$ and $y$.
4. Use $k$ to solve for unknown values of $y$ or $x$.

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Practice Example:

Problem:
If $y = 6$ when $x = 3$, find $y$ when $x = 9$.

1. First, find the constant $k$: $k = x \cdot y = 3 \cdot 6 = 18$

2. Now, use $k = 18$ to find $y$ when $x = 9$: $y = \frac{k}{x} = \frac{18}{9} = 2$

So, when $x = 9$, $y = 2$.

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5 Related Questions to Explore:

1. What happens if the constant $k$ is negative in inverse proportionality?
2. How do you graph inverse proportionality on a coordinate plane?
3. Can you apply inverse proportionality to real-world situations like physics or economics?
4. How do you solve for $x$ if $y$ and $k$ are known?
5. What is the difference between direct and inverse proportionality?

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Tip:

When working with inverse proportionality, always remember that the product of $x$ and $y$ is constant. If the values of $x$ and $y$ change, their product will remain the same.