Problem 45 - Algebra 2

Question

Each pair of values is from an inverse variation. Find the missing value. $$ (4,6),(x, 3) $$

Step-by-Step Solution

Answer

The missing value in the pair is \(x = 8\)

Step 1: Understand the Problem

Here it's mentioned that the pair of values are from an inverse variation. Which means, the product of values in each pair is constant.

Step 2: Calculate the Constant

Calculate the constant using the first pair (4,6) by multiplying them according to inverse variation rule. It comes as: \(k = 4 \times 6 = 24\). So, the constant \(k\) is 24.

Step 3: Find the Missing Value

Now, Use the calculated constant \(k\) and the given second pair \((x, 3)\) to find the missing value. This can be calculated by algebraic manipulation: \[x = k / 3\]. Thus, \(x = 24 / 3\)

Step 4: Final Calculation

On performing the final calculation, we conclude that \(x = 8\)

Key Concepts

Algebraic Manipulation

When dealing with inverse variation problems, algebraic manipulation is a key technique to solve for unknown values. An inverse variation is described by the equation \(xy = k\), where \(x\) and \(y\) are variables, and \(k\) is a constant. This means if one variable increases, the other decreases so that their product remains constant.

To find a missing value in such an equation, you must rearrange the equation to solve for the unknown variable. Suppose you know the value of \(y\) and need to find \(x\). You can manipulate the equation as follows:

• Start from the inverse variation formula: \(xy = k\).
• If \(k\) is known, divide both sides by \(y\), leading to \(x = \frac{k}{y}\).
• This manipulation provides a straightforward calculation for the missing variable \(x\).

Algebraic manipulation simplifies this process and allows you to swiftly calculate unknowns in problems involving inverse variation.

Constant Product

The concept of a constant product is foundational to understanding inverse variation. In inverse variation, no matter the values of \(x\) and \(y\), their product \(xy\) will always equal some constant \(k\). As a result, when you have a known pair, you can use it to establish this constant \(k\).

For example, if you are given a pair like \((4, 6)\), you can find the constant product by multiplying these values:

• Take the first value from the pair: 4
• Multiply it by the second value: 6
• The result, 24, is your constant \(k\) (i.e., \(4 \times 6 = 24\)).

With the constant \(k\) determined, it becomes a crucial piece of your puzzle, enabling you to solve for unknown values while maintaining the balance of the inverse variation relationship. Always remember that the essence of inverse variation lies in this constant product.

Missing Value Calculation

To calculate the missing value in an inverse variation problem, you'll utilize both the constant \(k\) and the known values from the equation. Once you've determined that \(xy = k\), and you've found a constant using given values, you can solve for your unknown.

Let's say you need to find \(x\) in the pair \((x, 3)\). Since you have already calculated \(k = 24\), you can find the unknown value as follows:

• Recall that for inverse variation \(xy = k\).
• Substitute the known value and constant: \(3x = 24\).
• To find \(x\), solve the equation by dividing both sides by 3: \(x = \frac{24}{3}\).
• Calculate to find \(x = 8\).

This method ensures that you isolate and solve for the missing value using logical steps. The process demonstrates the power of using a constant product in calculating unknowns and emphasizes the reliability of the inverse variation concept.