Question

When \(x=4, y=6 .\) For the given type of variation, find an equation that relates \(x\) and \(y .\) Then find the value of \(y\) when \(x=8\). \(x\) and \(y\) vary inversely.

Short answer

The equation for the inverse variation is \(y = 24/x\). When 'x' is 8, 'y' is 3.

Step-by-step solution

Determine the constant of variation 'k'

To find the value of k in the equation of inverse variation \(y = k/x\), you multiply the given 'x' and 'y' values. In this case, multiply 4 and 6 to get \(k = 4 \times 6 = 24\).

Write the equation for the inverse variation

Substitute the value of 'k' into the inverse variation equation \(y = k/x\). The equation then becomes \(y = 24/x\).

Find the value of 'y' when 'x' = 8

To find 'y' when 'x' is 8, substitute '8' in place of 'x' in the equation from Step 2: \(y = 24/8\). Compute this division to find 'y'.

Key concepts

Algebraic Equations

In mathematics, an algebraic equation is a statement of an equality containing one or more variables. These equations are like puzzles, where we need to find the value of the variable that makes the equation true.

Considering our example, the inverse variation is expressed through the algebraic equation \(y = \frac{k}{x}\). This specific type of equation represents a scenario where two variables (in this case, \(x\) and \(y\)) are linked in such a way that as one increases, the other decreases proportionally. The algebraic equation serves as a model that we can use to predict or understand how these variables interact. For example, if you know the value of \(x\) as we do in the exercise, you can rearrange the equation to solve for \(y\), and vice versa, always ensuring the product of \(x\) and \(y\) equals the constant \(k\).

Constant of Variation

The constant of variation, denoted as \(k\) in algebra, is the unchanging number in the relationship between two variables that vary inversely. In our exercise, once we discover \(k\) by multiplying the given values of \(x\) and \(y\) when they are provided, we lock in the tight rope that binds them together inversely.

The concept of the constant of variation is crucial because it encapsulates the entire relationship in a single number. In other words, knowing \(k\) lets us predict \(y\) for any \(x\), as long as the inverse relationship remains true. It's a powerful snapshot of how two quantities are entwined, allowing us to understand and express their interdependence quantitatively. Finding \(k\) is often the first step in analyzing inverse variation since it's the foundation we'll use to build our understanding and solve further problems.

Proportional Relationships

A proportional relationship is one where two quantities increase or decrease in tandem, at a constant rate. In our context, we discuss inverse proportionality: as one quantity increases, the other decreases proportionally, so their product is always the same.

This kind of relationship is described by an equation of the form \(y = \frac{k}{x}\), with \(k\) being the constant of proportionality. Understanding proportional relationships can help in predicting one variable when the other is known. For example, if we double the \(x\) value in our inverse variation equation, we can expect the \(y\) value to halve, ensuring the product \(xy\) remains equal to \(k\). This consistent predictability makes proportional relationships a fundamental concept in algebra that applies to various scientific laws and real-world problems, like calculating speed, density, and electrical resistance.