Problem 27 - Algebra 2

Question

Write each equation in the form \(y=\frac{k}{x}\). \(y=\frac{1}{2 x}\)

Step-by-Step Solution

Answer

The equation in the form \(y=\frac{k}{x}\) is \(y = \frac{1/2}{x}\)

Step 1: Identifying the constant of variation

From \(y = \frac{1}{2x}\), we can see that the constant of variation 'k' is 1/2.

Step 2: Rewriting the equation in the form \(y = \frac{k}{x}\)

Now, substitute 'k' with 1/2 in the form to get \(y = \frac{1/2}{x}\)

Key Concepts

Constant of Variation

The constant of variation, often represented as 'k', is a crucial part of any inverse variation equation. In an inverse variation, as one variable increases, the other decreases, and their product is constant. This constant, 'k', remains unchanged throughout the relationship.

For example, in the equation form of \( y = \frac{k}{x} \), 'k' determines the specific curve of the graph that this equation will produce. If 'k' is larger, the curve will be steeper, and if 'k' is smaller, the curve will be gentler.

In the example provided, \( y = \frac{1}{2x} \), the constant of variation is identified as 1/2. This indicates that for any chosen value of 'x', multiplying it with '2y' will result in the constant 1. Understanding 'k' helps in predicting how changes in 'x' affect 'y'.

Remember that the constant of variation can greatly influence the behavior of the equation and gives a simple way to understand the relationship between the variables.

Equation Transformation

Equation transformation is a powerful tool in algebra, allowing us to rearrange equations and better understand the relationships they describe. Transforming equations involves manipulating them while maintaining their underlying truths.

In this exercise, we transformed the equation \( y = \frac{1}{2x} \) into the standard form \( y = \frac{k}{x} \). To do this, we identified the constant 'k' as 1/2 and rewrote the equation by substituting 'k' accordingly. This transformation is crucial in understanding the direct relationship between 'y', 'x', and 'k'.

Steps typically involved in equation transformation include:

• Identifying key components or constants.
• Substituting known values.
• Rearranging terms to fit a desired formula.

Transforming equations helps simplify complex expressions and is a stepping stone in solving mathematical problems and modeling real-world scenarios.

Rational Expressions

Rational expressions are fractions wherein both the numerator and the denominator are polynomials. They are foundational to understanding relationships in mathematics, especially in the context of variation equations.

The equation \( y = \frac{1}{2x} \) is a perfect example of a rational expression, where the numerator (1) and the denominator (2x) are polynomial parts of the expression. This specific configuration fits into the context of inverse variation, illustrating how 'y' changes with different values of 'x'.

Key features of rational expressions include:

• Simplifying fractions by reducing common factors.
• Understanding that division by zero is undefined, meaning 'x' cannot be zero in our expression.
• Recognizing how changing numerator or denominator affects the overall value.

Grasping rational expressions allows students to handle more complex algebraic problems with confidence and prepares them for advanced topics in mathematics.