Problem 20 - Algebra A Combined Function

Question

If \(y\) varies inversely as \(x\), find the constant of variation and the inverse variation equation for each situation. \(y=0.6\) when \(x=0.3\)

Step-by-Step Solution

Answer

Constant of variation is 0.18; equation is \( y = \frac{0.18}{x} \).

Step 1: Understand Inverse Variation

Inverse variation means as one variable increases, the other decreases and vice versa. The relationship can be expressed as \( y = \frac{k}{x} \), where \(k\) is the constant of variation.

Step 2: Set Up the Initial Equation

We are given that \( y = 0.6 \) when \( x = 0.3 \). Using the inverse variation formula, substitute these values to find the constant \( k \): \[ 0.6 = \frac{k}{0.3} \].

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

Multiply both sides of the equation from Step 2 by 0.3 to solve for \(k\): \[ k = 0.6 \times 0.3 = 0.18 \]. This is the constant of variation.

Step 4: Write the Inverse Variation Equation

With \( k = 0.18 \), substitute back into the inverse variation formula to get the equation: \( y = \frac{0.18}{x} \). This is the inverse variation equation.

Key Concepts

Constant of Variation

In mathematics, particularly when dealing with inverse variation, the constant of variation is a crucial element. It allows us to understand how two variables relate inversely to each other. Simply put, in an inverse variation situation, as one variable increases, the other decreases proportionately.
The concept is often described using the formula:

• \[ y = \frac{k}{x} \]

where \(k\) represents the constant of variation.For example, in the problem where \(y = 0.6\) and \(x = 0.3\), we find that \(k\) is calculated as follows:

• Set up the equation: \(0.6 = \frac{k}{0.3}\).
• Solving this, \(k = 0.6 \times 0.3 = 0.18\).

Thus, \(k = 0.18\) in this specific case. The constant \(k\) helps in formulating the inverse relationship between \(y\) and \(x\). As you will notice, this constant helps define the inverse variation equation which can be used to find either variable if the other is known.

Inverse Variation Equation

The inverse variation equation is a fundamental concept in algebraic relationships where two quantities change in opposite directions. When one variable increases, the other one decreases, and this behavior can be algebraically represented.The general form of the inverse variation equation is:

• \[ y = \frac{k}{x} \]

Here, \(k\) is the constant of variation, as discussed earlier.When given specific values, such as in our example where \(y = 0.6\) and \(x = 0.3\), we found that \(k = 0.18\). Once we identified the value of \(k\), we plugged it back into the general formula to establish the inverse variation equation for this particular scenario:

• \[ y = \frac{0.18}{x} \]

With this equation, if you know one variable, you can easily find the other, showcasing the practical utility of the inverse equation in various real-world and mathematical contexts.

Algebraic Concepts

Inverse variation is just one of many fascinating algebraic concepts. Algebra uses symbols and letters to represent numbers and quantities in formulas and equations. Understanding algebraic concepts like inverse variation enhances our ability to model and solve problems involving relationships between variables. Key to this is grasping how changes in one variable affect another. Here, the inverse relationship shows that if one variable doubles, the other halves, maintaining the constant of variation as fixed. Connecting to the core principles, algebraic manipulation involves:

• Substituting known values into an equation.
• Solving for an unknown variable through operations like multiplication or division.

These steps align with standard algebraic practices, allowing the identification of unknown constants and the formulation of predictive equations such as the inverse variation equation. Truly appreciating these algebraic foundations can significantly advance your mathematical problem-solving toolkit.