Problem 16 - Algebra A Combined Function

Question

If \(y\) varies inversely as \(x\), find the constant of variation and the inverse variation equation for each situation. \(y=63\) when \(x=3\)

Step-by-Step Solution

Answer

The constant of variation is 189; the equation is \(y = \frac{189}{x}\).

Step 1: Understanding Inverse Variation

When two variables, say \(y\) and \(x\), vary inversely, it means their product is a constant. In mathematical terms, this relationship is expressed as \(xy = k\), where \(k\) is known as the constant of variation.

Step 2: Identify Known Values

From the problem, we know that \(y = 63\) when \(x = 3\). These values can be plugged into the inverse variation equation to find the constant \(k\).

Step 3: Find the Constant of Variation

Substitute the known values into the equation \(xy = k\):\[ 63 \times 3 = k \] Calculate this to find \(k\):\[ k = 189 \] So, the constant of variation \(k\) is 189.

Step 4: Write the Inverse Variation Equation

Now that we know \(k = 189\), we can write the inverse variation equation as \(xy = 189\). Alternatively, we can express \(y\) in terms of \(x\):\[ y = \frac{189}{x} \] This equation represents the inverse relationship between \(y\) and \(x\).

Key Concepts

Constant of Variation

Let's begin by unpacking the concept of "constant of variation". In an inverse variation, the constant of variation, often denoted as \(k\), is a fixed number that the product of the two variables, \(x\) and \(y\), equals. Essentially, it's the glue in the inverse relationship. Unlike direct variation, where one variable increases as the other increases, in inverse variation, when one variable increases, the other decreases, but their product remains consistent.
In our exercise, we are presented with values, \(y = 63\) and \(x = 3\). By substituting these values into the inverse variation equation \(xy = k\), we find that \(63 \times 3 = k \). Thus, \(k = 189\). This value, 189, will not change regardless of variations in \(x\) or \(y\), as long as the inverse relationship holds.

Mathematical Relationship

Inverse variation highlights a special kind of mathematical relationship between two variables. Here, as one quantity increases, the other one decreases proportionally to keep their product constant. This is fundamentally different from direct variation, where both variables move in the same direction.
To visualize this, think of a see-saw: as one side goes up, the other must come down to maintain balance. The equation \(xy = k\) exemplifies this relationship, showing that despite changes in \(x\) or \(y\), their interplay maintains \(k\) as a constant.
This relationship is vital in various real-world applications, like speed and travel time, where maintaining a constant product (distance) is key.

Inverse Variation Equation

The inverse variation equation is at the heart of understanding these types of relationships. It is expressed mathematically as \(xy = k\). For any set of variables \(x\) and \(y\) exhibiting inverse variation, this equation holds true.
In our example, with \(k = 189\), we write \(xy = 189\) or alternatively, express \(y\) solely in terms of \(x\) as \(y = \frac{189}{x}\).
This is convenient because it allows us to determine \(y\) for any given \(x\) without repeatedly solving new equations. This handy formula highlights not just the flexibility in calculations, but also underscores the efficiency that comes with understanding inverse variation: knowing the constant \(k\) simplifies everything.