Problem 14 - Elementary Algebra

Question

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=4, y=5\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=10\).

Step-by-Step Solution

Answer

a. \(k=20\). b. \(y=\frac{20}{x}\). c. \(y=2\) when \(x=10\).

Step 1: Understand inverse variation

In an inverse variation, the product of the two variables is constant. This means that for two variables, x and y, where the relationship is inverse, we can write: \(x \times y = k\) where \(k\) is the constant of proportionality.

Step 2: Finding the constant of proportionality, \(k\)

Given that when \(x=4\), \(y=5\), we can use these values to find \(k\). Using the inverse variation equation \(x \times y = k\), we have: \(4 \times 5 = k\) \(k = 20\).

Step 3: Write the inverse variation equation

Now that we have the constant \(k=20\), we can write the equation representing this inverse variation: \(x \times y = 20\) Alternatively, the equation can be written as: \(y = \frac{20}{x}\).

Step 4: Find \(y\) when \(x=10\)

Given the inverse variation equation \(y = \frac{20}{x}\), substitute \(x=10\) to find \(y\): \(y = \frac{20}{10}\) \(y = 2\).

Key Concepts

constant of proportionality

In the context of inverse variation, understanding the constant of proportionality is crucial. This constant is symbolized by the letter \(k\) and it plays a key role in defining the relationship between the variables \(x\) and \(y\). For inverse variation, the product of these two variables remains constant.

To illustrate this, consider the equation used for inverse variation: \(x \times y = k\). Here, \(k\) is the constant of proportionality. The value of \(k\) remains the same, provided the relation between \(x\) and \(y\) is inverse.

Let's say, for example, when \(x=4\) and \(y=5\), the value of \(k\) is found as follows:

\(k = x \times y = 4 \times 5 = 20\)

This tells us that no matter what values \(x\) and \(y\) take, as long as their product is 20, the relationship between them is inverse.

Understanding this constant \(k\) helps in predicting the relationship between \(x\) and \(y\) when one of the variables changes.

inverse variation equation

With the constant of proportionality \(k\) identified, we can now write down the inverse variation equation. The standard form of the equation is:\(x \times y = k\),
where \(k\) is the constant. Alternatively, this equation can be rearranged to solve for one of the variables: \(y = \frac{k}{x}\).

In the exercise given, we found \(k\) to be 20. So, plugging this value of \(k\) into the equation, we get: \(x \times y = 20\) or \(y = \frac{20}{x}\).

This equation tells us that for any value of \(x\), \(y\) can be found by dividing 20 by \(x\).

• For \(x = 4\), \(y = \frac{20}{4} = 5\).
• For \(x = 10\), \(y = \frac{20}{10} = 2\).

This inverse relationship ensures that as one variable increases, the other decreases to keep the product constant.

finding y

Once we have the inverse variation equation, finding the value of one variable when the other is known becomes straightforward. Let's focus on finding \(y\). Given the equation \(y = \frac{20}{x}\), we can determine \(y\) for any value of \(x\).

For example, if \(x = 10\), we can find \(y\) as follows:

\(y = \frac{20}{10} = 2\).
This simple calculation demonstrates how we can use the equation to find \(y\) for any given value of \(x\), maintaining the inverse proportionality relationship.

It's crucial to remember:

• The product of \(x\) and \(y\) always equals the constant \(k\).
• If \(x\) increases, \(y\) will decrease maintaining the product constant.
• This relationship only holds true for an inverse variation, not for other types of proportionalities.

Understanding how to manipulate this equation to find unknown values is a foundational skill in working with inverse variations.