Problem 30 - Algebra 2

Question

Each ordered pair is from an inverse variation. Find the constant of variation. $$ (0.9,4) $$

Step-by-Step Solution

Answer

The constant of variation is 3.6.

Step 1: Understanding the Inverse Variation Formula

An inverse variation can be represented by the equation \(y = k/x\). The constant of variation \(k\) is the constant (non-variable) term that remains the same for all pairs in an inverse variation.

Step 2: Substitute given ordered pair into the Formula

We substitute the given values from the ordered pair (0.9,4) into the equation. Here, x-value is 0.9 and y-value is 4. The equation becomes: 4 = k/0.9.

Step 3: Calculate the Constant of Variation

We solve this equation to find value of \(k\). Multiply each side of equation by 0.9 to isolate \(k\): \(k = 4 * 0.9 = 3.6\).

Key Concepts

Ordered Pairs

In mathematics, ordered pairs are used to denote relationships between two numbers. An ordered pair is represented as \(x, y\), where \(x\) is the first component and \(y\) is the second component. This pair shows how two sets of data are related to each other, allowing you to plot them on a graph where \(x\) is the horizontal axis and \(y\) is the vertical axis.
Ordered pairs are essential for understanding graphs, relationships, and functions. In inverse variation, these pairs help you identify the relationship that each component has with the constant of variation.

• They determine positions on a Cartesian plane.
• Used to describe functions, mappings, or relationships.
• Essential in understanding inverse and direct variations.

Constant of Variation

In an inverse variation, the constant of variation, often denoted as \(k\), plays a crucial role. It is the value that remains unchanged across all ordered pairs in the relationship. The equation for inverse variation is \(y = k/x\). This means for any value of \(x\), multiplying it by the corresponding \(y\), gives you the constant \(k\).
Finding \(k\) involves substituting the ordered pair into the inverse variation formula and solving for \(k\). For instance, given \(x = 0.9\) and \(y = 4\), the equation becomes \(4 = k/0.9\). Solving \(4 \times 0.9\) gives \(k = 3.6\).

• The constant \(k\) remains the same in all ordered pairs that share the inverse relationship.
• It acts as a unifying factor in equations involving inverse variations.
• Helps in understanding how different variables proportionally relate by inversely impacting one another.

Algebraic Equations

Algebraic equations are expressions that include variables and constants, linked by operators such as addition, subtraction, multiplication, and division. In the context of inverse variation, the algebraic equation is used to establish the relationship between two variables. The basic form here is \(y = k/x\), where \(y\) and \(x\) are variables, and \(k\) is the constant of variation.
Manipulating such equations involves substituting known values, simplifying, and solving for the unknown variable. This process requires understanding how to transform the equation, isolate the variable you're solving for, and carry out appropriate mathematical operations.

• Equations link variables in a structured way, offering solutions upon proper manipulation.
• They're pivotal for expressing mathematical relationships and solving for unknowns.
• In inverse variations, algebraic equations help find the constant or understand the relationship trends.