Problem 44 - Algebra 2

Question

Each pair of values is from an inverse variation. Find the missing value. $$ (2,5),(4, y) $$

Step-by-Step Solution

Answer

The missing value in the pair (4, y) is 2.5.

Step 1: Identify the known values

For the given inverse variation pairs, the known values are \(x_1 = 2\), \(y_1 = 5\) and \(x_2 = 4\). The unknown value that needs to be found is \(y_2\).

Step 2: Understand the formula for inverse variation

The formula for inverse variation is \(x_1y_1 = x_2y_2\).

Step 3: Substitute the known values into the formula

When we substitute the known values into the formula, we get \(2 * 5 = 4 * y_2\), which simplifies to \(10 = 4 * y_2\).

Step 4: Solve for the unknown value

We now need to solve the equation for \(y_2\). We can do this by dividing both sides of the equation by 4, which gives \(y_2 = 10 / 4 = 2.5\).

Key Concepts

Understanding a Pair of Values in Inverse Variation

In the context of inverse variation, a pair of values refers to two related quantities that behave according to an inverse relationship. If one value increases, the other decreases proportionally, and vice versa. This relationship can be expressed mathematically by the equation \(x_1y_1 = k\), where \(k\) is a constant.

For example, when given a pair of values like \((2, 5)\), it indicates that when \(x\) is 2, \(y\) is 5, and their product is a constant. In solving problems involving inverse variation, it's essential to recognize these pairs as they help establish the governing equation.

• Each pair follows the rule of inverse variation.
• The product of the values in a pair remains constant.

Understanding how these pairs work is crucial for setting up and solving the equations that determine unknowns.

Identifying the Missing Value

In problems involving inverse variation, like the one presented, identifying the missing value involves determining which part of the pair isn't given. It’s the number we are tasked with finding to complete the expression of the variation.

Suppose in our example, we know the complete pair \((2, 5)\) and the partial pair \((4, y)\). Here, \(y\) is the missing value that we need to find.

• Start by noting down what values you have.
• Write down what you need to find to complete the set.

Recognizing which value is missing helps direct the problem-solving process, making it easier to set up the necessary equation for finding it.

Determining the Unknown Value

The term "unknown value" is used to describe a quantity we don't know yet but need to find, often represented by a variable like \(y\) in the given task. To determine this unknown within an inverse variation context, you would use the equation formed by the pairs.

In our problem, the unknown \(y_2\) is from the pair \((4, y)\). This means when \(x\) is 4, \(y\) is determined by the relationship with the first pair \((2, 5)\). Since we know the equation \(x_1y_1 = x_2y_2\), the steps to find the unknown are clearly laid out.

• Identify your known values and equation.
• Substitute them into the equation.

This structured approach provides clarity and step-by-step guidance to solve for the unknown.

Effective Equation Solving for Inverse Variation

Equation solving in the context of inverse variation involves rearranging and substituting values to isolate the unknown. The steps are straightforward, following the pattern of inputting known values into the inverse variation formula \(x_1y_1 = x_2y_2\).

Here's a recap with our problem:

• Start with \(2 \times 5 = 4 \times y\).
• Simplify this to \(10 = 4y\).
• Divide both sides by 4 to solve: \(y = 2.5\).

The key to solving these equations is ensuring you substitute correctly and consistently across each step to maintain the integrity of the mathematical relationship. Breaking the equation into simpler parts often helps highlight necessary actions and solutions.