Problem 30 - Algebra 2

Question

Write each equation in the form \(y=\frac{k}{x}\). \(x y=-0.01\)

Step-by-Step Solution

Answer

The equation in the form \(y=\frac{k}{x}\) is \(y=\frac{-0.01}{x}\).

Step 1: Understanding the original equation

The original equation is \(x y=-0.01\). In this equation, \(x\) and \(y\) are variables and -0.01 is a constant.

Step 2: Isolate y

In order to isolate \(y\) on one side of the equation, divide both sides of the equation by \(x\). This gives: \(y=\frac{-0.01}{x}\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is an essential skill in solving equations effectively. It involves rearranging terms, performing operations like addition, subtraction, multiplication, and division to isolate variables or make the equation easier to solve. In our original equation, we have: - An equation in the form of \( xy = -0.01 \) To express \( y \) in terms of \( x \), we aim to isolate \( y \). This means getting \( y \) alone on one side of the equation. Instruction suggests dividing both sides by \( x \), which is a common algebraic technique:- Performing the operation: \( \frac{xy}{x} = \frac{-0.01}{x} \) simplifies to \( y = \frac{-0.01}{x} \). If you're new to these operations, remember that whatever you do to one side of the equation, you must do to the other side to maintain equality. This results in transforming the original form to the desired inverse variation form.

Constant of Variation

The concept of the constant of variation is crucial in inverse variation problems. In our context, when we reformulate our equation to \( y = \frac{k}{x} \), the \( k \) represents a constant. - In the given exercise, the constant of variation is \(-0.01\). The significance of this constant lies in its role in describing how \( x \) and \( y \) are related. Since \( k \) remains unchanged as \( x \) and \( y \) vary, it provides invaluable information about the relationship's strength and direction. Specifically, since \( k \) is negative, it indicates an inverse relationship where an increase in \( x \) leads to a decrease in \( y \) and vice versa.Understanding constants is fundamental because they define the specific condition or rule governing how variables interact.

Rational Equations

Rational equations involve one or more fractions whose numerators and/or denominators contain variables. In the reformulated equation \( y = \frac{-0.01}{x} \), we recognize it as a rational equation. Here are some important features about these equations:- They often take the form \( \frac{a}{b} = c \), where \( a \), \( b \), and \( c \) include variables and constants. In an inverse variation, the relationship between the two variables tends to follow a specific pattern seen in rational expressions, where the product of the variables remains constant.- Something key to remember: only for a specific non-zero \( x \), \( y \) will have a valid value; division by zero is undefined in mathematics.Grasping how to navigate rational equations will enable you to accurately solve relations involving variable dependencies, as they frequently appear in various scientific and mathematical fields.